INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III:
CANONICAL
SPLITTINGS
OF
THE
LOG-THETA-LATTICE
Shinichi
Mochizuki
May
2020
Abstract.
The
present
paper
constitutes
the
third
paper
in
a
series
of
four
papers
and
may
be
regarded
as
the
culmination
of
the
abstract
conceptual
por-
tion
of
the
theory
developed
in
the
series.
In
the
present
paper,
we
study
the
theory
surrounding
the
log-theta-lattice,
a
highly
non-commutative
two-dimensional
dia-
gram
of
“miniature
models
of
conventional
scheme
theory”,
called
Θ
±ell
NF-Hodge
theaters.
Here,
we
recall
that
Θ
±ell
NF-Hodge
theaters
were
associated,
in
the
first
paper
of
the
series,
to
certain
data,
called
initial
Θ-data,
that
includes
an
elliptic
curve
E
F
over
a
number
field
F
,
together
with
a
prime
number
l
≥
5.
Each
ar-
row
of
the
log-theta-lattice
corresponds
to
a
certain
gluing
operation
between
the
Θ
±ell
NF-Hodge
theaters
in
the
domain
and
codomain
of
the
arrow.
The
horizontal
arrows
of
the
log-theta-lattice
are
defined
as
certain
versions
of
the
“Θ-link”
that
was
constructed,
in
the
second
paper
of
the
series,
by
applying
the
theory
of
Hodge-
Arakelov-theoretic
evaluation
—
i.e.,
evaluation
in
the
style
of
the
scheme-theoretic
Hodge-Arakelov
theory
established
by
the
author
in
previous
papers
—
of
the
[reciprocal
of
the
l-th
root
of
the]
theta
function
at
l
-torsion
points.
In
the
present
paper,
we
focus
on
the
theory
surrounding
the
log-link
between
Θ
±ell
NF-
Hodge
theaters.
The
log-link
is
obtained,
roughly
speaking,
by
applying,
at
each
[say,
for
simplicity,
nonarchimedean]
valuation
of
the
number
field
under
consider-
ation,
the
local
p-adic
logarithm.
The
significance
of
the
log-link
lies
in
the
fact
that
it
allows
one
to
construct
log-shells,
i.e.,
roughly
speaking,
slightly
adjusted
forms
of
the
image
of
the
local
units
at
the
valuation
under
consideration
via
the
local
p-adic
logarithm.
The
theory
of
log-shells
was
studied
extensively
in
a
previ-
ous
paper
by
the
author.
The
vertical
arrows
of
the
log-theta-lattice
are
given
by
the
log-link.
Consideration
of
various
properties
of
the
log-theta-lattice
leads
natu-
rally
to
the
establishment
of
multiradial
algorithms
for
constructing
“splitting
monoids
of
logarithmic
Gaussian
procession
monoids”.
Here,
we
recall
that
“multiradial
algorithms”
are
algorithms
that
make
sense
from
the
point
of
view
of
an
“alien
arithmetic
holomorphic
structure”,
i.e.,
the
ring/scheme
structure
of
a
Θ
±ell
NF-Hodge
theater
related
to
a
given
Θ
±ell
NF-Hodge
theater
by
means
of
a
non-ring/scheme-theoretic
horizontal
arrow
of
the
log-theta-lattice.
These
loga-
rithmic
Gaussian
procession
monoids,
or
LGP-monoids,
for
short,
may
be
thought
of
as
the
log-shell-theoretic
versions
of
the
Gaussian
monoids
that
were
studied
in
the
second
paper
of
the
series.
Finally,
by
applying
these
multiradial
algorithms
for
splitting
monoids
of
LGP-monoids,
we
obtain
estimates
for
the
log-volume
of
these
LGP-monoids.
Explicit
computations
of
these
estimates
will
be
applied,
in
the
fourth
paper
of
the
series,
to
derive
various
diophantine
results.
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
Contents:
Introduction
§0.
Notations
and
Conventions
§1.
The
Log-theta-lattice
§2.
Multiradial
Theta
Monoids
§3.
Multiradial
Logarithmic
Gaussian
Procession
Monoids
Introduction
In
the
following
discussion,
we
shall
continue
to
use
the
notation
of
the
In-
troduction
to
the
first
paper
of
the
present
series
of
papers
[cf.
[IUTchI],
§I1].
In
particular,
we
assume
that
are
given
an
elliptic
curve
E
F
over
a
number
field
F
,
to-
gether
with
a
prime
number
l
≥
5.
In
the
first
paper
of
the
series,
we
introduced
and
studied
the
basic
properties
of
Θ
±ell
NF-Hodge
theaters,
which
may
be
thought
of
as
miniature
models
of
the
conventional
scheme
theory
surrounding
the
given
elliptic
curve
E
F
over
the
number
field
F
.
In
the
present
paper,
which
forms
the
third
paper
of
the
series,
we
study
the
theory
surrounding
the
log-link
between
Θ
±ell
NF-Hodge
theaters.
The
log-link
induces
an
isomorphism
between
the
underlying
D-Θ
±ell
NF-
Hodge
theaters
and,
roughly
speaking,
is
obtained
by
applying,
at
each
[say,
for
simplicity,
nonarchimedean]
valuation
v
∈
V,
the
local
p
v
-adic
logarithm
to
the
lo-
cal
units
[cf.
Proposition
1.3,
(i)].
The
significance
of
the
log-link
lies
in
the
fact
that
it
allows
one
to
construct
log-shells,
i.e.,
roughly
speaking,
slightly
adjusted
forms
of
the
image
of
the
local
units
at
v
∈
V
via
the
local
p
v
-adic
logarithm.
The
theory
of
log-shells
was
studied
extensively
in
[AbsTopIII].
The
introduction
of
log-shells
leads
naturally
to
the
construction
of
new
versions
—
namely,
the
×μ
×μ
-/Θ
×μ
Θ
×μ
gau
-links
studied
LGP
-/Θ
lgp
-links
[cf.
Definition
3.8,
(ii)]
—
of
the
Θ-/Θ
in
[IUTchI],
[IUTchII].
The
resulting
[highly
non-commutative!]
diagram
of
iterates
×μ
×μ
of
the
log-
[i.e.,
the
vertical
arrows]
and
Θ
×μ
-/Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-links
[i.e.,
the
horizontal
arrows]
—
which
we
refer
to
as
the
log-theta-lattice
[cf.
Definitions
1.4;
3.8,
(iii),
as
well
as
Fig.
I.1
below,
in
the
case
of
the
Θ
×μ
LGP
-link]
—
plays
a
central
role
in
the
theory
of
the
present
series
of
papers.
..
.
⏐
log
⏐
...
...
Θ
×μ
LGP
n,m+1
Θ
×μ
LGP
n,m
−→
−→
±ell
HT
Θ
⏐
log
⏐
±ell
..
.
⏐
log
⏐
NF
HT
Θ
NF
⏐
log
⏐
..
.
Θ
×μ
LGP
n+1,m+1
Θ
×μ
LGP
n+1,m
−→
−→
HT
Θ
⏐
log
⏐
±ell
±ell
HT
Θ
⏐
log
⏐
NF
NF
..
.
Fig.
I.1:
The
[LGP-Gaussian]
log-theta-lattice
Θ
×μ
LGP
−→
Θ
×μ
LGP
−→
...
...
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
3
Consideration
of
various
properties
of
the
log-theta-lattice
leads
naturally
to
the
establishment
of
multiradial
algorithms
for
constructing
“splitting
monoids
of
logarithmic
Gaussian
procession
monoids”
[cf.
Theorem
A
below].
Here,
we
recall
that
“multiradial
algorithms”
[cf.
the
discussion
of
[IUTchII],
Introduc-
tion]
are
algorithms
that
make
sense
from
the
point
of
view
of
an
“alien
arithmetic
holomorphic
structure”,
i.e.,
the
ring/scheme
structure
of
a
Θ
±ell
NF-Hodge
theater
related
to
a
given
Θ
±ell
NF-Hodge
theater
by
means
of
a
non-ring/scheme-
×μ
×μ
theoretic
Θ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-link.
These
logarithmic
Gaussian
procession
monoids,
or
LGP-monoids,
for
short,
may
be
thought
of
as
the
log-shell-theoretic
versions
of
the
Gaussian
monoids
that
were
studied
in
[IUTchII].
Finally,
by
apply-
ing
these
multiradial
algorithms
for
splitting
monoids
of
LGP-monoids,
we
obtain
estimates
for
the
log-volume
of
these
LGP-monoids
[cf.
Theorem
B
below].
These
estimates
will
be
applied
to
verify
various
diophantine
results
in
[IUTchIV].
Recall
[cf.
[IUTchI],
§I1]
the
notion
of
an
F-prime-strip.
An
F-prime-strip
consists
of
data
indexed
by
the
valuations
v
∈
V;
roughly
speaking,
the
data
at
each
v
consists
of
a
Frobenioid,
i.e.,
in
essence,
a
system
of
monoids
over
a
base
category.
For
instance,
at
v
∈
V
bad
,
this
data
may
be
thought
of
as
an
isomorphic
copy
of
the
monoid
with
Galois
action
Π
v
O
F
v
—
where
we
recall
that
O
F
denotes
the
multiplicative
monoid
of
nonzero
integral
v
elements
of
the
completion
of
an
algebraic
closure
F
of
F
at
a
valuation
lying
over
v
[cf.
[IUTchI],
§I1,
for
more
details].
The
p
v
-adic
logarithm
log
v
:
O
F
×
→
F
v
at
v
v
then
defines
a
natural
Π
v
-equivariant
isomorphism
of
ind-topological
modules
∼
∼
(O
F
×μ
⊗
Q
→
)
O
F
×
⊗
Q
→
F
v
v
v
—
where
we
recall
the
notation
“O
F
×μ
=
O
F
×
/O
F
μ
”
from
the
discussion
of
[IUTchI],
v
v
v
§1
—
which
allows
one
to
equip
O
F
×
⊗
Q
with
the
field
structure
arising
from
the
v
field
structure
of
F
v
.
The
portion
at
v
of
the
log-link
associated
to
an
F-prime-strip
[cf.
Definition
1.1,
(iii);
Proposition
1.2]
may
be
thought
of
as
the
correspondence
log
Π
v
O
F
−→
Π
v
O
F
v
v
in
which
one
thinks
of
the
copy
of
“O
F
”
on
the
right
as
obtained
from
the
field
v
structure
induced
by
the
p
v
-adic
logarithm
on
the
tensor
product
with
Q
of
the
copy
of
the
units
“O
F
×
⊆
O
F
”
on
the
left.
Since
this
correspondence
induces
an
v
v
isomorphism
of
topological
groups
between
the
copies
of
Π
v
on
either
side,
one
may
think
of
Π
v
as
“immune
to”/“neutral
with
respect
to”
—
or,
in
the
terminology
of
the
present
series
of
papers,
“coric”
with
respect
to
—
the
transformation
constituted
by
the
log-link.
This
situation
is
studied
in
detail
in
[AbsTopIII],
§3,
and
reviewed
in
Proposition
1.2
of
the
present
paper.
By
applying
various
results
from
absolute
anabelian
geometry,
one
may
algorithmically
reconstruct
a
copy
of
the
data
“Π
v
O
F
”
from
Π
v
.
Moreover,
v
4
SHINICHI
MOCHIZUKI
by
applying
Kummer
theory,
one
obtains
natural
isomorphisms
between
this
“coric
version”
of
the
data
“Π
v
O
F
”
and
the
copies
of
this
data
that
appear
on
v
either
side
of
the
log-link.
On
the
other
hand,
one
verifies
immediately
that
these
Kummer
isomorphisms
are
not
compatible
with
the
coricity
of
the
copy
of
the
data
“Π
v
O
F
”
algorithmically
constructed
from
Π
v
.
This
phenomenon
is,
in
v
some
sense,
the
central
theme
of
the
theory
of
[AbsTopIII],
§3,
and
is
reviewed
in
Proposition
1.2,
(iv),
of
the
present
paper.
The
introduction
of
the
log-link
leads
naturally
to
the
construction
of
log-
shells
at
each
v
∈
V.
If,
for
simplicity,
v
∈
V
bad
,
then
the
log-shell
at
v
is
given,
roughly
speaking,
by
the
compact
additive
module
×
I
v
=
p
−1
v
·
log
v
(O
K
v
)
⊆
K
v
⊆
F
v
def
[cf.
Definition
1.1,
(i),
(ii);
Remark
1.2.2,
(i),
(ii)].
One
has
natural
functorial
algorithms
for
constructing
various
versions
of
the
notion
of
a
log-shell
—
i.e.,
mono-analytic/holomorphic
and
étale-like/Frobenius-like
—
from
D
-/D-
/F
-/F-prime-strips
[cf.
Proposition
1.2,
(v),
(vi),
(vii),
(viii),
(ix)].
Although,
as
discussed
above,
the
relevant
Kummer
isomorphisms
are
not
compatible
with
the
log-link
“at
the
level
of
elements”,
the
log-shell
I
v
at
v
satisfies
the
important
property
×
⊆
I
v
;
log
v
(O
K
)
⊆
I
v
O
K
v
v
—
i.e.,
it
contains
the
images
of
the
Kummer
isomorphisms
associated
to
both
the
domain
and
the
codomain
of
the
log-link
[cf.
Proposition
1.2,
(v);
Remark
1.2.2,
(i),
(ii)].
In
light
of
the
compatibility
of
the
log-link
with
log-volumes
[cf.
Propositions
1.2,
(iii);
3.9,
(iv)],
this
property
will
ultimately
lead
to
upper
bounds
—
i.e.,
as
opposed
to
“precise
equalities”
—
in
the
computation
of
log-volumes
in
Corollary
3.12
[cf.
Theorem
B
below].
Put
another
way,
although
iterates
[cf.
Remark
1.1.1]
of
the
log-link
fail
to
be
compatible
with
the
various
Kummer
isomorphisms
that
arise,
one
may
nevertheless
consider
the
entire
diagram
that
results
from
considering
such
iterates
of
the
log-link
and
related
Kummer
isomorphisms
[cf.
Proposition
1.2,
(x)].
We
shall
refer
to
such
diagrams
...
→
...
•
→
•
↓
→
•
→
...
...
◦
—
i.e.,
where
the
horizontal
arrows
correspond
to
the
log-links
[that
is
to
say,
to
the
vertical
arrows
of
the
log-theta-lattice!];
the
“•’s”
correspond
to
the
Frobenioid-
theoretic
data
within
a
Θ
±ell
NF-Hodge
theater;
the
“◦”
corresponds
to
the
coric
version
of
this
data
[that
is
to
say,
in
the
terminology
discussed
below,
verti-
cally
coric
data
of
the
log-theta-lattice];
the
vertical/diagonal
arrows
correspond
to
the
various
Kummer
isomorphisms
—
as
log-Kummer
correspondences
[cf.
Theorem
3.11,
(ii);
Theorem
A,
(ii),
below].
Then
the
inclusions
of
the
above
display
may
be
interpreted
as
a
sort
of
“upper
semi-commutativity”
of
such
diagrams
[cf.
Remark
1.2.2,
(iii)],
which
we
shall
also
refer
to
as
the
“upper
semi-
compatibility”
of
the
log-link
with
the
relevant
Kummer
isomorphisms
—
cf.
the
discussion
of
the
“indeterminacy”
(Ind3)
in
Theorem
3.11,
(ii).
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
5
By
considering
the
log-links
associated
to
the
various
F-prime-strips
that
occur
in
a
Θ
±ell
NF-Hodge
theater,
one
obtains
the
notion
of
a
log-link
between
Θ
±ell
NF-
Hodge
theaters
†
HT
Θ
±ell
NF
log
−→
‡
HT
Θ
±ell
NF
[cf.
Proposition
1.3,
(i)].
As
discussed
above,
by
considering
the
iterates
of
the
log-
×μ
×μ
[i.e.,
the
vertical
arrows]
and
Θ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-links
[i.e.,
the
horizontal
arrows],
one
obtains
a
diagram
which
we
refer
to
as
the
log-theta-lattice
[cf.
Definitions
1.4;
3.8,
(iii),
as
well
as
Fig.
I.1,
in
the
case
of
the
Θ
×μ
LGP
-link].
As
discussed
above,
this
diagram
is
highly
noncommutative,
since
the
definition
of
the
log-link
depends,
in
an
essential
way,
on
both
the
additive
and
the
multiplicative
structures
—
i.e.,
on
the
ring
structure
—
of
the
various
local
rings
at
v
∈
V,
×μ
×μ
structures
which
are
not
preserved
by
the
Θ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-links
[cf.
Remark
1.4.1,
(i)].
So
far,
in
the
Introductions
to
[IUTchI],
[IUTchII],
as
well
as
in
the
present
Introduction,
we
have
discussed
various
“coricity”
properties
—
i.e.,
properties
of
invariance
with
respect
to
various
types
of
“transformations”
—
in
the
×μ
×μ
context
of
Θ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-links,
as
well
as
in
the
context
of
log-links.
In
the
context
of
the
log-theta-lattice,
it
becomes
necessary
to
distinguish
between
various
types
of
coricity.
That
is
to
say,
coricity
with
respect
to
log-links
[i.e.,
the
vertical
arrows
of
the
log-theta-lattice]
will
be
referred
to
as
vertical
coricity,
×μ
×μ
while
coricity
with
respect
to
Θ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-links
[i.e.,
the
horizontal
arrows
of
the
log-theta-lattice]
will
be
referred
to
as
horizontal
coricity.
On
the
other
hand,
coricity
properties
that
hold
with
respect
to
all
of
the
arrows
of
the
log-theta-lattice
will
be
referred
to
as
bi-coricity
properties.
Relative
to
the
analogy
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory
[cf.
[IUTchI],
§I4],
we
recall
that
a
Θ
±ell
NF-Hodge
the-
ater,
which
may
be
thought
of
as
a
miniature
model
of
the
conventional
scheme
theory
surrounding
the
given
elliptic
curve
E
F
over
the
number
field
F
,
corresponds
to
the
positive
characteristic
scheme
theory
surrounding
a
hyperbolic
curve
over
a
positive
characteristic
perfect
field
that
is
equipped
with
a
nilpotent
ordinary
indige-
nous
bundle
[cf.
Fig.
I.2
below].
Then
the
rotation,
or
“juggling”,
effected
by
the
log-link
of
the
additive
and
multiplicative
structures
of
the
conventional
scheme
theory
represented
by
a
Θ
±ell
NF-Hodge
theater
may
be
thought
of
as
correspond-
ing
to
the
Frobenius
morphism
in
positive
characteristic
[cf.
the
discussion
of
[AbsTopIII],
§I1,
§I3,
§I5].
Thus,
just
as
the
Frobenius
morphism
is
completely
well-
defined
in
positive
characteristic,
the
log-link
may
be
thought
of
as
a
phenomenon
that
occurs
within
a
single
arithmetic
holomorphic
structure,
i.e.,
a
vertical
line
of
the
log-theta-lattice.
By
contrast,
the
essentially
non-ring/scheme-theoretic
relationship
between
Θ
±ell
NF-Hodge
theaters
constituted
by
the
Θ-/Θ
×μ
-/Θ
×μ
gau
-
×μ
×μ
n
/Θ
LGP
-/Θ
lgp
-links
corresponds
to
the
relationship
between
the
“mod
p
”
and
“mod
p
n+1
”
portions
of
the
ring
of
Witt
vectors,
in
the
context
of
a
canonical
lifting
of
the
original
positive
characteristic
data
[cf.
the
discussion
of
Remark
1.4.1,
(iii);
Fig.
I.2
below].
Thus,
the
log-theta-lattice,
taken
as
a
whole,
may
be
thought
of
as
corresponding
to
the
canonical
lifting
of
the
original
positive
characteristic
data,
equipped
with
a
corresponding
canonical
Frobenius
action/lifting
[cf.
Fig.
I.2
below].
Finally,
the
non-commutativity
of
the
log-theta-lattice
may
be
thought
of
as
corresponding
to
the
complicated
“intertwining”
that
occurs
in
the
theory
of
Witt
vectors
and
canonical
liftings
between
the
Frobenius
morphism
in
positive
6
SHINICHI
MOCHIZUKI
characteristic
and
the
mixed
characteristic
nature
of
the
ring
of
Witt
vectors
[cf.
the
discussion
of
Remark
1.4.1,
(ii),
(iii)].
One
important
consequence
of
this
“noncommutative
intertwining”
of
the
two
dimensions
of
the
log-theta-lattice
is
the
following.
Since
each
horizontal
arrow
×μ
×μ
of
the
log-theta-lattice
[i.e.,
the
Θ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-link]
may
only
be
used
to
relate
—
i.e.,
via
various
Frobenioids
—
the
multiplicative
portions
of
the
ring
structures
in
the
domain
and
codomain
of
the
arrow,
one
natural
approach
to
relating
the
additive
portions
of
these
ring
structures
is
to
apply
the
theory
of
log-shells.
That
is
to
say,
since
each
horizontal
arrow
is
compatible
with
the
canonical
splittings
[up
to
roots
of
unity]
discussed
in
[IUTchII],
Introduction,
of
the
theta/Gaussian
monoids
in
the
domain
of
the
horizontal
arrow
into
unit
group
and
value
group
portions,
it
is
natural
to
attempt
to
relate
the
ring
structures
on
either
side
of
the
horizontal
arrow
by
applying
the
canonical
splittings
to
·
relate
the
multiplicative
structures
on
either
side
of
the
horizontal
arrow
by
means
of
the
value
group
portions
of
the
theta/Gaussian
monoids;
·
relate
the
additive
structures
on
either
side
of
the
horizontal
arrow
by
means
of
the
unit
group
portions
of
the
theta/Gaussian
monoids,
shifted
once
via
a
vertical
arrow,
i.e.,
the
log-link,
so
as
to
“render
additive”
the
[a
priori]
multiplicative
structure
of
these
unit
group
portions.
Indeed,
this
is
the
approach
that
will
ultimately
be
taken
in
Theorem
3.11
[cf.
Theorem
A
below]
to
relating
the
ring
structures
on
either
side
of
a
horizontal
arrow.
On
the
other
hand,
in
order
to
actually
implement
this
approach,
it
will
be
necessary
to
overcome
numerous
technical
obstacles.
Perhaps
the
most
immediately
obvious
such
obstacle
lies
in
the
observation
[cf.
the
discussion
of
Remark
1.4.1,
(ii)]
that,
precisely
because
of
the
“noncommutative
intertwining”
nature
of
the
log-theta-lattice,
any
sort
of
algorithmic
construction
concerning
objects
lying
in
the
do-
main
of
a
horizontal
arrow
that
involves
vertical
shifts
[e.g.,
such
as
the
approach
to
relating
additive
structures
in
the
fashion
described
above]
cannot
be
“translated”
in
any
immediate
sense
into
an
algorithm
that
makes
sense
from
the
point
of
view
of
the
codomain
of
the
horizontal
arrow.
In
a
word,
our
approach
to
overcoming
this
technical
obstacle
consists
of
working
with
objects
in
the
vertical
line
of
the
log-theta-lattice
that
contains
the
domain
of
the
horizontal
arrow
under
consideration
that
satisfy
the
crucial
property
of
being
invariant
with
respect
to
vertical
shifts
—
i.e.,
shifts
via
iterates
of
the
log-link
[cf.
the
discussion
of
Remarks
1.2.2,
(iii);
1.4.1,
(ii)].
For
instance,
étale-like
objects
that
are
vertically
coric
satisfy
this
invariance
property.
On
the
other
hand,
as
discussed
in
the
beginning
of
[IUTchII],
Introduction,
in
the
theory
of
the
present
series
of
papers,
it
is
of
crucial
impor-
tance
to
be
able
to
relate
corresponding
Frobenius-like
and
étale-like
structures
to
one
another
via
Kummer
theory.
In
particular,
in
order
to
obtain
structures
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
7
that
are
invariant
with
respect
to
vertical
shifts,
it
is
necessary
to
consider
log-
Kummer
correspondences,
as
discussed
above.
Moreover,
in
the
context
of
such
log-Kummer
correspondences,
typically,
one
may
only
obtain
structures
that
are
invariant
with
respect
to
vertical
shifts
if
one
is
willing
to
admit
some
sort
of
in-
determinacy,
e.g.,
such
as
the
“upper
semi-compatibility”
[cf.
the
discussion
of
the
“indeterminacy”
(Ind3)
in
Theorem
3.11,
(ii)]
discussed
above.
Inter-universal
Teichmüller
theory
p-adic
Teichmüller
theory
number
field
F
hyperbolic
curve
C
over
a
positive
characteristic
perfect
field
[once-punctured]
elliptic
curve
X
over
F
nilpotent
ordinary
indigenous
bundle
P
over
C
Θ-link
arrows
of
the
log-theta-lattice
mixed
characteristic
extension
structure
of
a
ring
of
Witt
vectors
log-link
arrows
of
the
log-theta-lattice
the
Frobenius
morphism
in
positive
characteristic
the
entire
log-theta-lattice
the
resulting
canonical
lifting
+
canonical
Frobenius
action;
canonical
Frobenius
lifting
over
the
ordinary
locus
relatively
straightforward
original
construction
of
Θ
×μ
LGP
-link
relatively
straightforward
original
construction
of
canonical
liftings
highly
nontrivial
description
of
alien
arithmetic
holomorphic
structure
via
absolute
anabelian
geometry
highly
nontrivial
absolute
anabelian
reconstruction
of
canonical
liftings
Fig.
I.2:
Correspondence
between
inter-universal
Teichmüller
theory
and
p-adic
Teichmüller
theory
8
SHINICHI
MOCHIZUKI
One
important
property
of
the
log-link,
and
hence,
in
particular,
of
the
con-
struction
of
log-shells,
is
its
compatibility
with
the
F
±
l
-symmetry
discussed
in
the
Introductions
to
[IUTchI],
[IUTchII]
—
cf.
Remark
1.3.2.
Here,
we
recall
from
the
discussion
of
[IUTchII],
Introduction,
that
the
F
±
l
-symmetry
allows
one
to
relate
the
various
F-prime-strips
—
i.e.,
more
concretely,
the
various
copies
of
the
data
“Π
v
O
F
”
at
v
∈
V
bad
[and
their
analogues
for
v
∈
V
good
]
—
associated
v
to
the
various
labels
∈
F
l
that
appear
in
the
Hodge-Arakelov-theoretic
evaluation
of
[IUTchII]
in
a
fashion
that
is
compatible
with
·
the
distinct
nature
of
distinct
labels
∈
F
l
;
·
the
Kummer
isomorphisms
used
to
relate
Frobenius-like
and
étale-
like
versions
of
the
F-prime-strips
that
appear,
i.e.,
more
concretely,
the
various
copies
of
the
data
“Π
v
O
F
”
at
v
∈
V
bad
[and
their
analogues
v
for
v
∈
V
good
];
·
the
structure
of
the
underlying
D-prime-strips
that
appear,
i.e.,
more
concretely,
the
various
copies
of
the
[arithmetic]
tempered
fundamental
group
“Π
v
”
at
v
∈
V
bad
[and
their
analogues
for
v
∈
V
good
]
—
cf.
the
discussion
of
[IUTchII],
Introduction;
Remark
1.5.1;
Step
(vii)
of
the
proof
of
Corollary
3.12
of
the
present
paper.
This
compatibility
with
the
F
±
l
-symmetry
gives
rise
to
the
construction
of
·
vertically
coric
F
×μ
-prime-strips,
log-shells
by
means
of
the
arith-
metic
holomorphic
structures
under
consideration;
·
mono-analytic
F
×μ
-prime-strips,
log-shells
which
are
bi-coric
—
cf.
Theorem
1.5.
These
bi-coric
mono-analytic
log-shells
play
a
central
role
in
the
theory
of
the
present
paper.
One
notable
aspect
of
the
compatibility
of
the
log-link
with
the
F
±
l
-symmetry
in
the
context
of
the
theory
of
Hodge-Arakelov-theoretic
evaluation
developed
in
[IUTchII]
is
the
following.
One
important
property
of
mono-theta
environments
is
the
property
of
“isomorphism
class
compatibility”,
i.e.,
in
the
terminology
of
[EtTh],
“compatibility
with
the
topology
of
the
tempered
fundamental
group”
[cf.
the
discussion
of
Remark
2.1.1].
This
“isomorphism
class
compatibility”
allows
one
to
apply
the
Kummer
theory
of
mono-theta
environments
[i.e.,
the
theory
of
[EtTh]]
relative
to
the
ring-theoretic
basepoints
that
occur
on
either
side
of
the
log-link
[cf.
Remark
2.1.1,
(ii);
[IUTchII],
Remark
3.6.4,
(i)],
for
instance,
in
the
context
of
the
log-Kummer
correspondences
discussed
above.
Here,
we
recall
that
the
significance
of
working
with
such
“ring-theoretic
basepoints”
lies
in
the
fact
that
the
full
ring
structure
of
the
local
rings
involved
[i.e.,
as
opposed
to,
say,
just
the
multiplicative
portion
of
this
ring
structure]
is
necessary
in
order
to
construct
the
log-link.
That
is
to
say,
it
is
precisely
by
establishing
the
conjugate
synchronization
arising
from
the
F
±
l
-symmetry
relative
to
these
basepoints
that
occur
on
either
side
of
the
log-link
that
one
is
able
to
conclude
the
crucial
compatibility
of
this
conjugate
synchronization
with
the
log-link
discussed
in
Remark
1.3.2.
Thus,
in
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
9
summary,
one
important
consequence
of
the
“isomorphism
class
compatibility”
of
mono-theta
environments
is
the
simultaneous
compatibility
of
·
the
Kummer
theory
of
mono-theta
environments;
·
the
conjugate
synchronization
arising
from
the
F
±
l
-symmetry;
·
the
construction
of
the
log-link.
This
simultaneous
compatibility
is
necessary
in
order
to
perform
the
construction
of
the
[crucial!]
splitting
monoids
of
LGP-monoids
referred
to
above
—
cf.
the
discussion
of
Step
(vi)
of
the
proof
of
Corollary
3.12.
In
§2
of
the
present
paper,
we
continue
our
preparation
for
the
multiradial
con-
struction
of
splitting
monoids
of
LGP-monoids
given
in
§3
[of
the
present
paper]
by
presenting
a
global
formulation
of
the
essentially
local
theory
at
v
∈
V
bad
[cf.
[IUTchII],
§1,
§2,
§3]
concerning
the
interpretation,
via
the
notion
of
multiradial-
ity,
of
various
rigidity
properties
of
mono-theta
environments.
That
is
to
say,
although
much
of
the
[essentially
routine!]
task
of
formulating
the
local
theory
of
[IUTchII],
§1,
§2,
§3,
in
global
terms
was
accomplished
in
[IUTchII],
§4,
the
[again
essentially
routine!]
task
of
formulating
the
portion
of
this
local
theory
that
con-
cerns
multiradiality
was
not
explicitly
addressed
in
[IUTchII],
§4.
One
reason
for
this
lies
in
the
fact
that,
from
the
point
of
view
of
the
theory
to
be
developed
in
§3
of
the
present
paper,
this
global
formulation
of
multiradiality
properties
of
the
mono-
theta
environment
may
be
presented
most
naturally
in
the
framework
developed
in
§1
of
the
present
paper,
involving
the
log-theta-lattice
[cf.
Theorem
2.2;
Corollary
2.3].
Indeed,
the
étale-like
versions
of
the
mono-theta
environment,
as
well
as
the
various
objects
constructed
from
the
mono-theta
environment,
may
be
interpreted,
from
the
point
of
view
of
the
log-theta-lattice,
as
vertically
coric
structures,
and
are
Kummer-theoretically
related
to
their
Frobenius-like
[i.e.,
Frobenioid-
theoretic]
counterparts,
which
arise
from
the
[Frobenioid-theoretic
portions
of
the]
various
Θ
±ell
NF-Hodge
theaters
in
a
vertical
line
of
the
log-theta-lattice
[cf.
Theo-
rem
2.2,
(ii);
Corollary
2.3,
(ii),
(iii),
(iv)].
Moreover,
it
is
precisely
the
horizontal
×
-indeterminacies
acting
arrows
of
the
log-theta-lattice
that
give
rise
to
the
Z
×μ
on
copies
of
“O
”
that
play
a
prominent
role
in
the
local
multiradiality
theory
de-
veloped
in
[IUTchII]
[cf.
the
discussion
of
[IUTchII],
Introduction].
In
this
context,
it
is
useful
to
recall
from
the
discussion
of
[IUTchII],
Introduction
[cf.
also
Remark
2.2.1
of
the
present
paper],
that
the
essential
content
of
this
local
multiradiality
the-
ory
consists
of
the
observation
[cf.
Fig.
I.3
below]
that,
since
mono-theta-theoretic
cyclotomic
and
constant
multiple
rigidity
only
require
the
use
of
the
portion
of
O
F
×
,
for
v
∈
V
bad
,
given
by
the
torsion
subgroup
O
F
μ
v
v
⊆
O
F
×
[i.e.,
the
roots
of
unity],
v
the
triviality
of
the
composite
of
natural
morphisms
O
F
μ
v
→
O
F
×
v
O
F
×μ
v
has
the
effect
of
insulating
the
Kummer
theory
of
the
étale
theta
function
—
i.e.,
via
the
theory
of
the
mono-theta
environments
developed
in
[EtTh]
—
from
×
-indeterminacies
that
act
on
the
copies
of
“O
×μ
”
that
arise
in
the
F
×μ
-
the
Z
×μ
×μ
prime-strips
that
appear
in
the
Θ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-link.
10
SHINICHI
MOCHIZUKI
×
Z
id
O
F
μ
v
→
O
F
×μ
v
×
-indeterminacies
in
the
context
of
Fig.
I.3:
Insulation
from
Z
mono-theta-theoretic
cyclotomic,
constant
multiple
rigidity
In
§3
of
the
present
paper,
which,
in
some
sense,
constitutes
the
conclusion
of
the
theory
developed
thus
far
in
the
present
series
of
papers,
we
present
the
construction
of
the
[splitting
monoids
of]
LGP-monoids,
which
may
be
thought
of
as
a
multiradial
version
of
the
[splitting
monoids
of]
Gaussian
monoids
that
were
constructed
via
the
theory
of
Hodge-Arakelov-theoretic
evaluation
developed
in
[IUTchII].
In
order
to
achieve
this
multiradiality,
it
is
necessary
to
“multiradi-
alize”
the
various
components
of
the
construction
of
the
Gaussian
monoids
given
in
[IUTchII].
The
first
step
in
this
process
of
“multiradialization”
concerns
the
labels
j
∈
F
l
that
occur
in
the
Hodge-Arakelov-theoretic
evaluation
performed
in
[IUTchII].
That
is
to
say,
the
construction
of
these
labels,
together
with
the
closely
related
theory
of
F
l
-symmetry,
depend,
in
an
essential
way,
on
the
full
arithmetic
tempered
fundamental
groups
“Π
v
”
at
v
∈
V
bad
,
i.e.,
on
the
portion
of
the
arithmetic
holomorphic
structure
within
a
Θ
±ell
NF-Hodge
theater
which
is
not
shared
by
an
alien
arithmetic
holomorphic
structure
[i.e.,
an
arithmetic
holo-
morphic
structure
related
to
the
original
arithmetic
holomorphic
structure
via
a
horizontal
arrow
of
the
log-theta-lattice].
One
naive
approach
to
remedying
this
state
of
affairs
is
to
simply
consider
the
underlying
set,
of
cardinality
l
,
associated
to
F
l
,
which
we
regard
as
being
equipped
with
the
full
set
of
symmetries
given
by
arbitrary
permutation
automorphisms
of
this
underlying
set.
The
problem
with
this
approach
is
that
it
yields
a
situation
in
which,
for
each
label
j
∈
F
l
,
one
must
contend
with
an
indeterminacy
of
l
possibilities
for
the
element
of
this
underlying
set
that
corresponds
to
j
[cf.
[IUTchI],
Propositions
4.11,
(i);
6.9,
(i)].
From
the
point
of
view
of
the
log-volume
computations
to
be
performed
in
[IUTchIV],
this
degree
of
indeterminacy
gives
rise
to
log-volumes
which
are
“too
large”,
i.e.,
to
esti-
mates
that
are
not
sufficient
for
deriving
the
various
diophantine
results
obtained
in
[IUTchIV].
Thus,
we
consider
the
following
alternative
approach,
via
processions
[cf.
[IUTchI],
Propositions,
4.11,
6.9].
Instead
of
working
just
with
the
underlying
set
associated
to
F
l
,
we
consider
the
diagram
of
inclusions
of
finite
sets
S
±
1
→
S
±
1+1=2
→
...
→
S
±
j+1
→
...
→
S
±
1+l
=l
±
—
where
we
write
S
±
j+1
=
{0,
1,
.
.
.
,
j},
for
j
=
0,
.
.
.
,
l
,
and
we
think
of
each
of
these
finite
sets
as
being
subject
to
arbitrary
permutation
automorphisms.
That
is
to
say,
we
think
of
the
set
S
±
j+1
as
a
container
for
the
labels
0,
1,
.
.
.
,
j.
Thus,
for
each
j,
one
need
only
contend
with
an
indeterminacy
of
j
+
1
possibilities
for
the
element
of
this
container
that
corresponds
to
j.
In
particular,
if
one
allows
j
=
0,
.
.
.
,
l
to
vary,
then
this
approach
allows
one
to
reduce
the
resulting
label
±
indeterminacy
from
a
total
of
(l
±
)
l
possibilities
[where
we
write
l
±
=
1
+
l
=
def
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
11
(l+1)/2]
to
a
total
of
l
±
!
possibilities.
It
turns
out
that
this
reduction
will
yield
just
the
right
estimates
in
the
log-volume
computations
to
be
performed
in
[IUTchIV].
Moreover,
this
approach
satisfies
the
important
property
of
insulating
the
“core
label
0”
from
the
various
label
indeterminacies
that
occur.
Each
element
of
each
of
the
containers
S
±
j+1
may
be
thought
of
as
parametrizing
an
F-
or
D-prime-strip
that
occurs
in
the
Hodge-Arakelov-theoretic
evaluation
of
[IUTchII].
In
order
to
render
the
construction
multiradial,
it
is
necessary
to
replace
such
holomorphic
F-/D-prime-strips
by
mono-analytic
F
-/D
-prime-strips.
In
particular,
as
discussed
above,
one
may
construct,
for
each
such
F
-/D
-prime-
strip,
a
collection
of
log-shells
associated
to
the
various
v
∈
V.
Write
V
Q
for
the
set
of
valuations
of
Q.
Then,
in
order
to
obtain
objects
that
are
immune
to
the
various
label
indeterminacies
discussed
above,
we
consider,
for
each
element
∗
∈
S
±
j+1
,
and
for
each
[say,
for
simplicity,
nonarchimedean]
v
Q
∈
V
Q
,
·
the
direct
sum
of
the
log-shells
associated
to
the
prime-strip
labeled
by
the
given
element
∗
∈
S
±
j+1
at
the
v
∈
V
that
lie
over
v
Q
;
we
then
form
·
the
tensor
product,
over
the
elements
∗
∈
S
±
j+1
,
of
these
direct
sums.
This
collection
of
tensor
products
associated
to
v
Q
∈
V
Q
will
be
referred
to
as
the
tensor
packet
associated
to
the
collection
of
prime-strips
indexed
by
elements
of
S
±
j+1
.
One
may
carry
out
this
construction
of
the
tensor
packet
either
for
holomor-
phic
F-/D-prime-strips
[cf.
Proposition
3.1]
or
for
mono-analytic
F
-/D
-prime-
strips
[cf.
Proposition
3.2].
The
tensor
packets
associated
to
D
-prime-strips
will
play
a
crucial
role
in
the
theory
of
§3,
as
“multiradial
mono-analytic
containers”
for
the
principal
objects
of
interest
[cf.
the
discussion
of
Remark
3.12.2,
(ii)],
namely,
·
the
action
of
the
splitting
monoids
of
the
LGP-monoids
—
i.e.,
the
2
monoids
generated
by
the
theta
values
{q
j
}
j=1,...,l
—
on
the
portion
of
v
the
tensor
packets
just
defined
at
v
∈
V
bad
[cf.
Fig.
I.4
below;
Propositions
3.4,
3.5;
the
discussion
of
[IUTchII],
Introduction];
×
)
j
”
of
[the
multiplicative
monoid
of
nonzero
·
the
action
of
copies
“(F
mod
elements
of]
the
number
field
F
mod
labeled
by
j
=
1,
.
.
.
,
l
on
the
product,
over
v
Q
∈
V
Q
,
of
the
portion
of
the
tensor
packets
just
defined
at
v
Q
[cf.
Fig.
I.5
below;
Propositions
3.3,
3.7,
3.10].
/
±
S
±
1
→
/
±
/
±
S
±
1+1=2
2
2
q
1
q
j
→
...
→
/
±
/
±
.
.
.
/
±
S
±
j+1
q
(l
)
→
...
→
/
±
/
±
.
.
.
.
.
.
/
±
S
±
1+l
=l
±
Fig.
I.4:
Splitting
monoids
of
LGP-monoids
acting
on
tensor
packets
12
SHINICHI
MOCHIZUKI
×
(F
mod
)
1
/
±
S
±
1
→
/
±
/
±
×
(F
mod
)
j
→
.
.
.
S
±
1+1=2
→
/
±
/
±
.
.
.
/
±
×
(F
mod
)
l
→
.
.
.
→
/
±
/
±
.
.
.
.
.
.
/
±
S
±
j+1
S
±
1+l
=l
±
×
Fig.
I.5:
Copies
of
F
mod
acting
on
tensor
packets
×
Indeed,
these
[splitting
monoids
of]
LGP-monoids
and
copies
“(F
mod
)
j
”
of
[the
multiplicative
monoid
of
nonzero
elements
of]
the
number
field
F
mod
admit
nat-
ural
embeddings
into/actions
on
the
various
tensor
packets
associated
to
labeled
F-
±ell
prime-strips
in
each
Θ
±ell
NF-Hodge
theater
n,m
HT
Θ
NF
of
the
log-theta-lattice.
One
then
obtains
vertically
coric
versions
of
these
splitting
monoids
of
LGP-
×
)
j
”
of
[the
multiplicative
monoid
of
nonzero
monoids
and
labeled
copies
“(F
mod
elements
of]
the
number
field
F
mod
by
applying
suitable
Kummer
isomorphisms
between
·
log-shells/tensor
packets
associated
to
[labeled]
F-prime-strips
and
·
log-shells/tensor
packets
associated
to
[labeled]
D-prime-strips.
Finally,
by
passing
to
the
·
log-shells/tensor
packets
associated
to
[labeled]
D
-prime-strips
—
i.e.,
by
forgetting
the
arithmetic
holomorphic
structure
associated
to
a
specific
vertical
line
of
the
log-theta-lattice
—
one
obtains
the
desired
multiradial
representation,
i.e.,
description
in
terms
that
make
sense
from
the
point
of
view
of
an
alien
arithmetic
holomorphic
structure,
of
the
splitting
monoids
of
LGP-
monoids
and
labeled
copies
of
the
number
field
F
mod
discussed
above.
This
passage
to
the
multiradial
representation
is
obtained
by
admitting
the
following
three
types
of
indeterminacy:
(Ind1):
This
is
the
indeterminacy
that
arises
from
the
automorphisms
of
proces-
sions
of
D
-prime-strips
that
appear
in
the
multiradial
representation
—
i.e.,
more
concretely,
from
permutation
automorphisms
of
the
label
sets
S
±
j+1
that
appear
in
the
processions
discussed
above,
as
well
as
from
the
automorphisms
of
the
D
-prime-strips
that
appear
in
these
processions.
(Ind2):
This
is
the
[“non-(Ind1)
portion”
of
the]
indeterminacy
that
arises
from
the
automorphisms
of
the
F
×μ
-prime-strips
that
appear
in
the
Θ-/Θ
×μ
-
×μ
×μ
non
/Θ
×μ
,
gau
-/Θ
LGP
-/Θ
lgp
-link
—
i.e.,
in
particular,
at
[for
simplicity]
v
∈
V
×
-indeterminacies
acting
on
local
copies
of
“O
×μ
”
[cf.
the
above
the
Z
discussion].
(Ind3):
This
is
the
indeterminacy
that
arises
from
the
upper
semi-compatibility
of
the
log-Kummer
correspondences
associated
to
the
specific
vertical
line
of
the
log-theta-lattice
under
consideration
[cf.
the
above
discussion].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
13
A
detailed
description
of
this
multiradial
representation,
together
with
the
indeter-
minacies
(Ind1),
(Ind2)
is
given
in
Theorem
3.11,
(i)
[and
summarized
in
Theorem
A,
(i),
below;
cf.
also
Fig.
I.6
below].
q
1
q
q
j
2
(l
)
2
/
±
→
/
±
/
±
→
.
.
.
→
/
±
/
±
.
.
.
/
±
→
.
.
.
→
/
±
/
±
.
.
.
.
.
.
/
±
×
(F
mod
)
1
×
(F
mod
)
j
×
(F
mod
)
l
Fig.
I.6:
The
full
multiradial
representation
One
important
property
of
the
multiradial
representation
discussed
above
con-
cerns
the
relationship
between
the
three
main
components
—
i.e.,
roughly
speaking,
log-shells,
splitting
monoids
of
LGP-monoids,
and
number
fields
—
of
this
multira-
dial
representation
and
the
log-Kummer
correspondence
of
the
specific
vertical
line
of
the
log-theta-lattice
under
consideration.
This
property
—
which
may
be
thought
of
as
a
sort
of
“non-interference”,
or
“mutual
compatibility”,
prop-
erty
—
asserts
that
the
multiplicative
monoids
constituted
by
the
splitting
monoids
×
“do
not
interfere”,
relative
to
the
various
ar-
of
LGP-monoids
and
copies
of
F
mod
rows
that
occur
in
the
log-Kummer
correspondence,
with
the
local
units
at
v
∈
V
that
give
rise
to
the
log-shells.
In
the
case
of
splitting
monoids
of
LGP-monoids,
this
non-interference/mutual
compatibility
property
is,
in
essence,
a
formal
conse-
quence
of
the
existence
of
the
canonical
splittings
[up
to
roots
of
unity]
of
the
theta/Gaussian
monoids
that
appear
into
unit
group
and
value
group
portions
[cf.
the
discussion
of
[IUTchII],
Introduction].
Here,
we
recall
that,
in
the
case
of
the
theta
monoids,
these
canonical
splittings
are,
in
essence,
a
formal
consequence
of
the
constant
multiple
rigidity
property
of
mono-theta
environments
reviewed
above.
In
the
case
of
copies
of
F
mod
,
this
non-interference/mutual
compatibility
property
is,
in
essence,
a
formal
consequence
of
the
well-known
fact
in
elementary
algebraic
number
theory
that
any
nonzero
element
of
a
number
field
that
is
inte-
gral
at
every
valuation
of
the
number
field
is
necessarily
a
root
of
unity.
These
mutual
compatibility
properties
are
described
in
detail
in
Theorem
3.11,
(ii),
and
summarized
in
Theorem
A,
(ii),
below.
Another
important
property
of
the
multiradial
representation
discussed
above
concerns
the
relationship
between
the
three
main
components
—
i.e.,
roughly
speak-
ing,
log-shells,
splitting
monoids
of
LGP-monoids,
and
number
fields
—
of
this
multiradial
representation
and
the
Θ
×μ
LGP
-links,
i.e.,
the
horizontal
arrows
of
the
log-theta-lattice
under
consideration.
This
property
—
which
may
be
thought
of
as
a
property
of
compatibility
with
the
Θ
×μ
LGP
-link
—
asserts
that
the
cyclotomic
rigidity
isomorphisms
that
appear
in
the
Kummer
theory
surrounding
the
splitting
×
×
-indeterminacies
are
immune
to
the
Z
monoids
of
LGP-monoids
and
copies
of
F
mod
×μ
×μ
-prime-strips
that
appear
that
act
on
the
copies
of
“O
”
that
arise
in
the
F
×μ
in
the
Θ
LGP
-link.
In
the
case
of
splitting
monoids
of
LGP-monoids,
this
prop-
erty
amounts
precisely
to
the
multiradiality
theory
developed
in
§2
[cf.
the
above
14
SHINICHI
MOCHIZUKI
discussion],
i.e.,
in
essence,
to
the
mono-theta-theoretic
cyclotomic
rigidity
×
property
reviewed
in
the
above
discussion.
In
the
case
of
copies
of
F
mod
,
this
prop-
erty
follows
from
the
theory
surrounding
the
construction
of
the
cyclotomic
rigidity
isomorphisms
discussed
in
[IUTchI],
Example
5.1,
(v).
These
compatibility
prop-
erties
are
described
in
detail
in
Theorem
3.11,
(iii),
and
summarized
in
Theorem
A,
(iii),
below.
At
this
point,
we
pause
to
observe
that
although
considerable
attention
has
been
devoted
so
far
in
the
present
series
of
papers,
especially
in
[IUTchII],
to
the
theory
of
Gaussian
monoids,
not
so
much
attention
has
been
devoted
[i.e.,
outside
of
[IUTchI],
§5;
[IUTchII],
Corollaries
4.7,
4.8]
to
[the
multiplicative
monoids
×
×
.
These
copies
of
F
mod
enter
into
the
theory
of
the
constituted
by]
copies
of
F
mod
multiradial
representation
discussed
above
in
the
form
of
various
types
of
global
Frobenioids
in
the
following
way.
If
one
starts
from
the
number
field
F
mod
,
one
of
[stack-
natural
Frobenioid
that
can
be
associated
to
F
mod
is
the
Frobenioid
F
mod
theoretic]
arithmetic
line
bundles
on
[the
spectrum
of
the
ring
of
integers
of]
F
mod
discussed
in
[IUTchI],
Example
5.1,
(iii)
[cf.
also
Example
3.6
of
the
present
paper].
From
the
point
of
view
of
the
theory
surrounding
the
multiradial
representation
”:
discussed
above,
there
are
two
natural
ways
to
approach
the
construction
of
“F
mod
(
MOD
)
(Rational
Function
Torsor
Version):
This
approach
consists
of
con-
×
sidering
the
category
F
MOD
of
F
mod
-torsors
equipped
with
trivializations
at
each
v
∈
V
[cf.
Example
3.6,
(i),
for
more
details].
(
mod
)
(Local
Fractional
Ideal
Version):
This
approach
consists
of
consid-
ering
the
category
F
mod
of
collections
of
integral
structures
on
the
various
completions
K
v
at
v
∈
V
and
morphisms
between
such
collections
of
in-
×
tegral
structures
that
arise
from
multiplication
by
elements
of
F
mod
[cf.
Example
3.6,
(ii),
for
more
details].
Then
one
has
natural
isomorphisms
of
Frobenioids
F
mod
∼
→
F
MOD
∼
→
F
mod
×
×
×
that
induce
the
respective
identity
morphisms
F
mod
→
F
mod
→
F
mod
on
the
asso-
ciated
rational
function
monoids
[cf.
[FrdI],
Corollary
4.10].
In
particular,
at
first
glance,
F
MOD
and
F
mod
appear
to
be
“essentially
equivalent”
objects.
On
the
other
hand,
when
regarded
from
the
point
of
view
of
the
multiradial
representations
discussed
above,
these
two
constructions
exhibit
a
number
of
signif-
icant
differences
—
cf.
Fig.
I.7
below;
the
discussion
of
Remarks
3.6.2,
3.10.1.
For
instance,
whereas
the
construction
of
(
MOD
)
depends
only
on
the
multiplica-
×
tive
structure
of
F
mod
,
the
construction
of
(
mod
)
involves
the
module,
i.e.,
the
additive,
structure
of
the
localizations
K
v
.
The
global
portion
of
the
Θ
×μ
LGP
-link
×μ
(respectively,
the
Θ
lgp
-link)
is,
by
definition
[cf.
Definition
3.8,
(ii)],
constructed
by
means
of
the
realification
of
the
Frobenioid
that
appears
in
the
construction
of
(
MOD
)
(respectively,
(
mod
)).
This
means
that
the
construction
of
the
global
por-
tion
of
the
Θ
×μ
LGP
-link
—
which
is
the
version
of
the
Θ-link
that
is
in
fact
ultimately
used
in
the
theory
of
the
multiradial
representation
—
depends
only
on
the
multi-
×
,
together
with
the
various
valuation
plicative
monoid
structure
of
a
copy
of
F
mod
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
15
×
homomorphisms
F
mod
→
R
associated
to
v
∈
V.
Thus,
the
mutual
compatibility
×
[discussed
above]
of
copies
of
F
mod
with
the
log-Kummer
correspondence
implies
that
one
may
perform
this
construction
of
the
global
portion
of
the
Θ
×μ
LGP
-link
in
a
fashion
that
is
immune
to
the
“upper
semi-compatibility”
indeterminacy
(Ind3)
[discussed
above].
By
contrast,
the
construction
of
(
mod
)
involves
integral
struc-
tures
on
the
underlying
local
additive
modules
“K
v
”,
i.e.,
from
the
point
of
view
of
the
multiradial
representation,
integral
structures
on
log-shells
and
tensor
packets
of
log-shells,
which
are
subject
to
the
“upper
semi-compatibility”
indeterminacy
(Ind3)
[discussed
above].
In
particular,
the
log-Kummer
correspondence
subjects
the
construction
of
(
mod
)
to
“substantial
distortion”.
On
the
other
hand,
the
es-
sential
role
played
by
local
integral
structures
in
the
construction
of
(
mod
)
enables
one
to
compute
the
global
arithmetic
degree
of
the
arithmetic
line
bundles
consti-
”
in
terms
of
log-volumes
on
log-shells
tuted
by
objects
of
the
category
“F
mod
and
tensor
packets
of
log-shells
[cf.
Proposition
3.9,
(iii)].
This
property
of
the
construction
of
(
mod
)
will
play
a
crucial
role
in
deriving
the
explicit
estimates
for
such
log-volumes
that
are
obtained
in
Corollary
3.12
[cf.
Theorem
B
below].
F
MOD
F
mod
biased
toward
multiplicative
structures
biased
toward
additive
structures
easily
related
to
value
group/non-coric
portion
“(−)
”
of
Θ
×μ
LGP
-link
easily
related
to
unit
group/coric
×μ
portion
“(−)
×μ
”
of
Θ
×μ
LGP
-/Θ
lgp
-link,
i.e.,
mono-analytic
log-shells
admits
precise
log-Kummer
correspondence
only
admits
“upper
semi-compatible”
log-Kummer
correspondence
rigid,
but
not
suited
to
explicit
computation
subject
to
substantial
distortion,
but
suited
to
explicit
estimates
Fig.
I.7:
F
MOD
versus
F
mod
∼
Thus,
in
summary,
the
natural
isomorphism
F
MOD
→
F
mod
discussed
above
plays
the
important
role,
in
the
context
of
the
multiradial
representation
discussed
above,
of
relating
·
the
multiplicative
structure
of
the
global
number
field
F
mod
to
the
additive
structure
of
F
mod
,
16
SHINICHI
MOCHIZUKI
·
the
unit
group/coric
portion
“(−)
×μ
”
of
the
Θ
×μ
LGP
-link
to
the
value
×μ
group/non-coric
portion
“(−)
”
of
the
Θ
LGP
-link.
Finally,
in
Corollary
3.12
[cf.
also
Theorem
B
below],
we
apply
the
multiradial
representation
discussed
above
to
estimate
certain
log-volumes
as
follows.
We
begin
by
introducing
some
terminology
[cf.
Definition
3.8,
(i)].
We
shall
refer
to
the
object
that
arises
in
any
of
the
versions
[including
realifications]
of
the
global
Frobenioid
”
discussed
above
—
such
as,
for
instance,
the
global
realified
Frobenioid
“F
mod
×μ
×μ
that
occurs
in
the
codomain
of
the
Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-link
—
by
considering
the
arithmetic
divisor
determined
by
the
zero
locus
of
the
elements
“q
”
at
v
∈
V
bad
v
as
a
q-pilot
object.
The
log-volume
of
the
q-pilot
object
will
be
denoted
by
−
|log(q)|
∈
R
—
so
|log(q)|
>
0
[cf.
Corollary
3.12;
Theorem
B].
In
a
similar
vein,
we
shall
refer
to
the
object
that
arises
in
the
global
realified
Frobenioid
that
occurs
in
the
domain
of
×μ
×μ
the
Θ
×μ
gau
-/Θ
LGP
-/Θ
lgp
-link
by
considering
the
arithmetic
divisor
determined
by
the
2
zero
locus
of
the
collection
of
theta
values
“{q
j
}
j=1,...,l
”
at
v
∈
V
bad
as
a
Θ-pilot
v
object.
The
log-volume
of
the
holomorphic
hull
—
cf.
Remark
3.9.5,
(i);
Step
(xi)
of
the
proof
of
Corollary
3.12
—
of
the
union
of
the
collection
of
possible
images
of
the
Θ-pilot
object
in
the
multiradial
representation
—
i.e.,
where
we
recall
that
these
“possible
images”
are
subject
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
—
will
be
denoted
by
{+∞}
−
|log(Θ)|
∈
R
[cf.
Corollary
3.12;
Theorem
B].
Here,
the
reader
might
find
the
use
of
the
notation
“−”
and
“|
.
.
.
|”
confusing
[i.e.,
since
this
notation
suggests
that
−
|log(Θ)|
is
a
non-positive
real
number,
which
would
appear
to
imply
that
the
possibility
that
−
|log(Θ)|
=
+∞
may
be
excluded
from
the
outset].
The
reason
for
the
use
of
this
notation,
however,
is
to
express
the
point
of
view
that
−
|log(Θ)|
should
be
regarded
as
a
positive
real
multiple
of
−
|log(q)|
[i.e.,
which
is
indeed
a
negative
real
number!]
plus
a
possible
error
term,
which
[a
priori!]
might
be
equal
to
+∞.
Then
the
content
of
Corollary
3.12,
Theorem
B
may
be
summarized,
roughly
speaking
[cf.
Remark
3.12.1,
(ii)],
as
a
result
concerning
the
negativity
of
the
Θ-pilot
log-volume
|log(Θ)|
def
{−∞}.
Relative
to
—
i.e.,
where
we
write
|log(Θ)|
=
−(−
|log(Θ)|)
∈
R
the
analogy
between
the
theory
of
the
present
series
of
papers
and
complex/p-adic
Teichmüller
theory
[cf.
[IUTchI],
§I4],
this
result
may
be
thought
of
as
a
statement
to
the
effect
that
“the
pair
consisting
of
a
number
field
equipped
with
an
elliptic
curve
is
metrically
hyperbolic,
i.e.,
has
negative
curvature”.
That
is
to
say,
it
may
be
thought
of
as
a
sort
of
analogue
of
the
inequality
χ
S
=
−
dμ
S
<
0
S
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
17
arising
from
the
classical
Gauss-Bonnet
formula
on
a
hyperbolic
Riemann
sur-
face
of
finite
type
S
[where
we
write
χ
S
for
the
Euler
characteristic
of
S
and
dμ
S
for
the
Kähler
metric
on
S
determined
by
the
Poincaré
metric
on
the
upper
half-plane
—
cf.
the
discussion
of
Remark
3.12.3],
or,
alternatively,
of
the
inequality
(1
−
p)(2g
X
−
2)
≤
0
that
arises
by
computing
global
degrees
of
line
bundles
in
the
context
of
the
Hasse
invariant
that
arises
in
p-adic
Teichmüller
theory
[where
X
is
a
smooth,
proper
hyperbolic
curve
of
genus
g
X
over
the
ring
of
Witt
vectors
of
a
perfect
field
of
characteristic
p
which
is
canonical
in
the
sense
of
p-adic
Teichmüller
theory
—
cf.
the
discussion
of
Remark
3.12.4,
(v)].
The
proof
of
Corollary
3.12
[i.e.,
Theorem
B]
is
based
on
the
following
funda-
mental
observation:
the
multiradial
representation
discussed
above
yields
two
tautologically
equivalent
ways
to
compute
the
q-pilot
log-volume
−
|log(q)|
—
cf.
Fig.
I.8
below;
Step
(xi)
of
the
proof
of
Corollary
3.12.
That
is
to
say,
suppose
±ell
that
one
starts
with
the
q-pilot
object
in
the
Θ
±ell
NF-Hodge
theater
1,0
HT
Θ
NF
at
(1,
0),
which
we
think
of
as
being
represented,
via
the
approach
of
(
mod
),
by
means
of
the
action
of
the
various
q
,
for
v
∈
V
bad
,
on
the
log-shells
that
arise,
v
Θ
±ell
NF
log
±ell
Θ
×μ
LGP
±ell
HT
−→
1,0
HT
Θ
NF
,
from
the
various
local
“O
×μ
’s”
via
the
log-link
±ell
in
the
Θ
±ell
NF-Hodge
theater
1,−1
HT
Θ
NF
at
(1,
−1).
Thus,
if
one
considers
the
value
group
“(−)
”
and
unit
group
“(−)
×μ
”
portions
of
the
codomain
of
1,−1
±ell
0,0
HT
Θ
NF
the
Θ
×μ
LGP
-link
−→
1,0
HT
Θ
NF
in
the
context
of
the
arithmetic
holomorphic
structure
of
the
vertical
line
(1,
◦),
this
action
on
log-shells
may
be
thought
of
as
a
somewhat
intricate
“intertwining”
between
these
value
group
and
unit
group
portions
[cf.
Remark
3.12.2,
(ii)].
On
the
other
hand,
the
Θ
×μ
LGP
-
±ell
Θ
×μ
±ell
LGP
1,0
link
0,0
HT
Θ
NF
−→
HT
Θ
NF
constitutes
a
sort
of
gluing
isomorphism
between
the
arithmetic
holomorphic
structures
associated
to
the
vertical
lines
(0,
◦)
and
(1,
◦)
that
is
based
on
forgetting
this
intricate
intertwining,
i.e.,
by
working
solely
with
abstract
isomorphisms
of
F
×μ
-prime-strips.
Thus,
in
order
to
relate
the
arithmetic
holomorphic
structures,
say,
at
(0,
0)
and
(1,
0),
one
must
apply
the
multiradial
representation
discussed
above.
That
is
to
say,
one
starts
by
applying
the
theory
of
bi-coric
mono-analytic
log-shells
given
in
Theorem
1.5.
One
then
applies
the
Kummer
theory
surrounding
the
splitting
monoids
of
theta/Gaussian
monoids
and
copies
of
the
number
field
F
mod
,
which
allows
one
to
pass
from
the
Frobenius-like
versions
of
various
objects
that
appear
in
—
i.e.,
that
are
necessary
in
order
to
consider
—
the
Θ
×μ
LGP
-link
to
the
corresponding
étale-like
versions
of
these
objects
that
appear
in
the
multiradial
representation.
This
passage
from
Frobenius-like
versions
to
étale-like
versions
is
referred
to
as
the
operation
of
Kummer-detachment
[cf.
Fig.
I.8;
Remark
1.5.4,
(i)].
As
discussed
above,
this
operation
of
Kummer-detachment
is
possible
precisely
18
SHINICHI
MOCHIZUKI
as
a
consequence
of
the
compatibility
of
the
multiradial
representation
with
the
indeterminacies
(Ind1),
(Ind2),
(Ind3),
hence,
in
particular,
with
the
Θ
×μ
LGP
-link.
Here,
we
recall
that
since
the
log-theta-lattice
is,
as
discussed
above,
far
from
commutative,
in
order
to
represent
the
various
“log-link-conjugates”
at
(0,
m)
[for
m
∈
Z]
in
terms
that
may
be
understood
from
the
point
of
view
of
the
arith-
metic
holomorphic
structure
at
(1,
0),
one
must
work
[not
only
with
the
Kummer
isomorphisms
at
a
single
(0,
m),
but
rather]
with
the
entire
log-Kummer
corre-
spondence.
In
particular,
one
must
take
into
account
the
indeterminacy
(Ind3).
Once
one
completes
the
operation
of
Kummer-detachment
so
as
to
obtain
vertically
coric
versions
of
objects
on
the
vertical
line
(0,
◦),
one
then
passes
to
multiradial
objects,
i.e.,
to
the
“final
form”
of
the
multiradial
representation,
by
taking
into
account
[once
again]
the
indeterminacy
(Ind1),
i.e.,
that
arises
from
working
with
[mono-analytic!]
D
-
[as
opposed
to
D-!]
prime-strips.
Finally,
one
computes
the
log-volume
of
the
holomorphic
hull
of
this
“final
form”
multiradial
representation
of
the
Θ-pilot
object
—
i.e.,
subject
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)!
—
and
concludes
the
desired
estimates
from
the
tautological
observation
that
the
log-theta-lattice
—
and,
in
particular,
the
“gluing
isomorphism”
constituted
by
the
Θ
×μ
LGP
-link
—
were
constructed
precisely
in
such
a
way
as
to
ensure
that
the
computation
of
the
log-volume
of
the
holomorphic
hull
of
the
union
of
the
collection
of
possible
images
of
the
Θ-pilot
object
[cf.
the
definition
of
|log(Θ)|]
necessarily
amounts
to
a
computation
of
[an
upper
bound
for]
|log(q)|
multiradial
representation
at
0-column
(0,
◦)
Kummer-
detach-
ment
via
log-
Kummer
⇑
permutation
symmetry
of
≈
étale-picture
com-
pati-
bly
with
Θ
×μ
LGP
-
link
Θ-pilot
object
in
Θ
±ell
NF-Hodge
theater
at
(0,
0)
multiradial
representation
at
1-column
(1,
◦)
com-
pari-
son
via
(−)
-portion,
(−)
×μ
-portion
≈
×μ
of
Θ
LGP
-link
⇓
hol.
hull,
log-
vol.
q-pilot
object
in
Θ
±ell
NF-Hodge
theater
at
(1,
0)
Fig.
I.8:
Two
tautologically
equivalent
ways
to
compute
the
log-volume
of
the
q-pilot
object
at
(1,
0)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
19
—
cf.
Fig.
I.8;
Step
(xi)
of
the
proof
of
Corollary
3.12.
That
is
to
say,
the
“gluing
isomorphism”
constituted
by
the
Θ
×μ
LGP
-link
relates
two
distinct
“arithmetic
holo-
morphic
structures”,
i.e.,
two
distinct
copies
of
conventional
ring/scheme
theory,
that
are
glued
together
precisely
by
means
of
a
relation
that
identifies
the
Θ-pilot
object
in
the
domain
of
the
Θ
×μ
LGP
-link
with
the
q-pilot
object
in
the
codomain
of
the
×μ
Θ
LGP
-link.
Thus,
once
one
sets
up
such
an
apparatus,
the
computation
of
the
log-
volume
of
the
holomorphic
hull
of
the
union
of
possible
images
of
the
Θ-pilot
object
in
the
domain
of
the
Θ
×μ
LGP
-link
in
terms
of
the
q-pilot
object
in
the
codomain
of
×μ
the
Θ
LGP
-link
amounts
—
tautologically!
—
to
the
computation
of
the
log-volume
of
the
q-pilot
object
[in
the
codomain
of
the
Θ
×μ
LGP
-link]
in
terms
of
itself,
i.e.,
to
a
computation
that
reflects
certain
intrinsic
properties
of
this
q-pilot
object.
This
is
the
content
of
Corollary
3.12
[i.e.,
Theorem
B].
As
discussed
above,
this
sort
of
“computation
of
intrinsic
properties”
in
the
present
context
of
a
number
field
equipped
with
an
elliptic
curve
may
be
regarded
as
analogous
to
the
“computa-
tions
of
intrinsic
properties”
reviewed
above
in
the
classical
complex
and
p-adic
cases.
We
conclude
the
present
Introduction
with
the
following
summaries
of
the
main
results
of
the
present
paper.
Theorem
A.
(Multiradial
Algorithms
for
Logarithmic
Gaussian
Proces-
sion
Monoids)
Fix
a
collection
of
initial
Θ-data
(F
/F,
X
F
,
l,
C
K
,
V,
V
bad
mod
,
)
as
in
[IUTchI],
Definition
3.1.
Let
{
n,m
HT
Θ
±ell
NF
}
n,m∈Z
be
a
collection
of
distinct
Θ
±ell
NF-Hodge
theaters
[relative
to
the
given
initial
Θ-data]
—
which
we
think
of
as
arising
from
a
LGP-Gaussian
log-theta-lattice
[cf.
Definition
3.8,
(iii)].
For
each
n
∈
Z,
write
n,◦
±ell
HT
D-Θ
NF
for
the
D-Θ
±ell
NF-Hodge
theater
determined,
up
to
isomorphism,
by
the
various
±ell
n,m
HT
Θ
NF
,
where
m
∈
Z,
via
the
vertical
coricity
of
Theorem
1.5,
(i)
[cf.
Remark
3.8.2].
(i)
(Multiradial
Representation)
Write
n,◦
R
LGP
for
the
collection
of
data
consisting
of
(a)
tensor
packets
of
log-shells;
(b)
splitting
monoids
of
LGP-monoids
acting
on
the
tensor
packets
of
(a);
(c)
copies,
labeled
by
j
∈
F
l
,
of
[the
multiplicative
monoid
of
nonzero
ele-
ments
of
]
the
number
field
F
mod
acting
on
the
tensor
packets
of
(a)
20
SHINICHI
MOCHIZUKI
[cf.
Theorem
3.11,
(i),
(a),
(b),
(c),
for
more
details]
regarded
up
to
indetermi-
nacies
of
the
following
two
types:
(Ind1)
the
indeterminacies
induced
by
the
automorphisms
of
the
procession
of
D
-prime-strips
Prc(
n,◦
D
T
)
that
gives
rise
to
the
tensor
packets
of
(a);
(Ind2)
the
[“non-(Ind1)
portion”
of
the]
indeterminacies
that
arise
from
the
au-
tomorphisms
of
the
F
×μ
-prime-strips
that
appear
in
the
Θ
×μ
LGP
-link,
non
×
i.e.,
in
particular,
at
[for
simplicity]
v
∈
V
,
the
Z
-indeterminacies
acting
on
local
copies
of
“O
×μ
”
—
cf.
Theorem
3.11,
(i),
for
more
details.
Then
n,◦
R
LGP
may
be
constructed
via
an
algorithm
in
the
procession
of
D
-prime-strips
Prc(
n,◦
D
T
),
which
is
functo-
rial
with
respect
to
isomorphisms
of
processions
of
D
-prime-strips.
For
n,
n
∈
Z,
the
permutation
symmetries
of
the
étale-picture
discussed
in
[IUTchI],
Corollary
6.10,
(iii);
[IUTchII],
Corollary
4.11,
(ii),
(iii)
[cf.
also
Corollary
2.3,
(ii);
Remarks
2.3.2
and
3.8.2,
of
the
present
paper],
induce
compatible
poly-
isomorphisms
∼
Prc(
n,◦
D
T
)
→
Prc(
n
,◦
D
T
);
n,◦
∼
R
LGP
→
n
,◦
R
LGP
which
are,
moreover,
compatible
with
the
bi-coricity
poly-isomorphisms
n,◦
∼
D
0
→
n
,◦
D
0
of
Theorem
1.5,
(iii)
[cf.
also
[IUTchII],
Corollaries
4.10,
(iv);
4.11,
(i)].
(ii)
(log-Kummer
Correspondence)
For
n,
m
∈
Z,
the
inverses
of
the
Kummer
isomorphisms
associated
to
the
various
F-prime-strips
and
NF-
±ell
bridges
that
appear
in
the
Θ
±ell
NF-Hodge
theater
n,m
HT
Θ
NF
induce
“inverse
Kummer”
isomorphisms
between
the
vertically
coric
data
(a),
(b),
(c)
of
(i)
and
the
corresponding
Frobenioid-theoretic
data
arising
from
each
Θ
±ell
NF-
±ell
Hodge
theater
n,m
HT
Θ
NF
[cf.
Theorem
3.11,
(ii),
(a),
(b),
(c),
for
more
de-
tails].
Moreover,
as
one
varies
m
∈
Z,
the
corresponding
Kummer
isomor-
phisms
[i.e.,
inverses
of
“inverse
Kummer”
isomorphisms]
of
splitting
monoids
of
LGP-monoids
[cf.
(i),
(b)]
and
labeled
copies
of
the
number
field
F
mod
[cf.
(i),
(c)]
are
mutually
compatible,
relative
to
the
log-links
of
the
n-th
column
of
the
LGP-Gaussian
log-theta-lattice
under
consideration,
in
the
sense
that
the
only
portions
of
the
[Frobenioid-theoretic]
domains
of
these
Kummer
isomorphisms
that
are
possibly
related
to
one
another
via
the
log-links
consist
of
roots
of
unity
in
the
domains
of
the
log-links
[multiplication
by
which
corresponds,
via
the
log-link,
to
an
“addition
by
zero”
indeterminacy,
i.e.,
to
no
indeterminacy!]
—
cf.
Proposi-
tion
3.5,
(ii),
(c);
Proposition
3.10,
(ii);
Theorem
3.11,
(ii),
for
more
details.
On
the
other
hand,
the
Kummer
isomorphisms
of
tensor
packets
of
log-shells
[cf.
(i),
(a)]
are
subject
to
a
certain
“indeterminacy”
as
follows:
(Ind3)
as
one
varies
m
∈
Z,
these
Kummer
isomorphisms
of
tensor
packets
of
log-shells
are
“upper
semi-compatible”,
relative
to
the
log-links
of
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
21
n-th
column
of
the
LGP-Gaussian
log-theta-lattice
under
consideration,
in
and
a
sense
that
involves
certain
natural
inclusions
“⊆”
at
v
Q
∈
V
non
Q
—
cf.
Proposition
3.5,
certain
natural
surjections
“”
at
v
Q
∈
V
arc
Q
(ii),
(a),
(b);
Theorem
3.11,
(ii),
for
more
details.
Finally,
as
one
varies
m
∈
Z,
these
Kummer
isomorphisms
of
tensor
packets
of
log-shells
are
[precisely!]
compatible,
relative
to
the
log-links
of
the
n-th
column
of
the
LGP-Gaussian
log-theta-lattice
under
consideration,
with
the
respective
log-
volumes
[cf.
Proposition
3.9,
(iv)].
(iii)
(Θ
×μ
LGP
-Link
Compatibility)
The
various
Kummer
isomorphisms
of
(ii)
satisfy
compatibility
properties
with
the
various
horizontal
arrows
—
i.e.,
Θ
×μ
LGP
-
links
—
of
the
LGP-Gaussian
log-theta-lattice
under
consideration
as
follows:
The
tensor
packets
of
log-shells
[cf.
(i),
(a)]
are
compatible,
relative
to
the
relevant
Kummer
isomorphisms,
with
[the
unit
group
portion
“(−)
×μ
”
of
]
the
Θ
×μ
LGP
-link
[cf.
the
indeterminacy
“(Ind2)”
of
(i)];
we
refer
to
Theorem
3.11,
(iii),
(a),
(b),
for
more
details.
The
identity
automorphism
on
the
objects
that
appear
in
the
construction
of
the
splitting
monoids
of
LGP-monoids
via
mono-theta
envi-
ronments
[cf.
(i),
(b)]
is
compatible,
relative
to
the
relevant
Kummer
isomorphisms
and
isomorphisms
of
mono-theta
environments,
with
the
Θ
×μ
LGP
-link
[cf.
the
inde-
terminacy
“(Ind2)”
of
(i)];
we
refer
to
Theorem
3.11,
(iii),
(c),
for
more
details.
The
identity
automorphism
on
the
objects
that
appear
in
the
construction
of
the
labeled
copies
of
the
number
field
F
mod
[cf.
(i),
(c)]
is
compatible,
relative
to
the
relevant
Kummer
isomorphisms
and
cyclotomic
rigidity
isomorphisms
[cf.
the
discussion
of
Remark
2.3.2;
the
constructions
of
[IUTchI],
Example
5.1,
(v)],
with
the
Θ
×μ
LGP
-link
[cf.
the
indeterminacy
“(Ind2)”
of
(i)];
we
refer
to
Theorem
3.11,
(iii),
(d),
for
more
details.
Theorem
B.
(Log-volume
Estimates
for
Multiradially
Represented
Split-
ting
Monoids
of
Logarithmic
Gaussian
Procession
Monoids)
Suppose
that
we
are
in
the
situation
of
Theorem
A.
Write
−
|log(Θ)|
∈
R
{+∞}
for
the
procession-normalized
mono-analytic
log-volume
[where
the
average
is
taken
over
j
∈
F
l
—
cf.
Remark
3.1.1,
(ii),
(iii),
(iv);
Proposition
3.9,
(i),
(ii);
Theorem
3.11,
(i),
(a),
for
more
details]
of
the
holomorphic
hull
[cf.
Remark
3.9.5,
(i)]
of
the
union
of
the
possible
images
of
a
Θ-pilot
object
[cf.
Definition
3.8,
(i)],
relative
to
the
relevant
Kummer
isomorphisms
[cf.
Theorems
A,
(ii);
3.11,
(ii)],
in
the
multiradial
representation
of
Theorems
A,
(i);
3.11,
(i),
which
we
regard
as
subject
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
described
in
Theorems
A,
(i),
(ii);
3.11,
(i),
(ii).
Write
−
|log(q)|
∈
R
for
the
procession-normalized
mono-analytic
log-volume
of
the
image
of
a
q-pilot
object
[cf.
Definition
3.8,
(i)],
relative
to
the
relevant
Kummer
isomor-
phisms
[cf.
Theorems
A,
(ii);
3.11,
(ii)],
in
the
multiradial
representation
of
22
SHINICHI
MOCHIZUKI
Theorems
A,
(i);
3.11,
(i),
which
we
do
not
regard
as
subject
to
the
indetermina-
cies
(Ind1),
(Ind2),
(Ind3)
described
in
Theorems
A,
(i),
(ii);
3.11,
(i),
(ii).
Here,
we
recall
the
definition
of
the
symbol
“”
as
the
result
of
identifying
the
labels
“0”
and
“F
l
”
[cf.
[IUTchII],
Corollary
4.10,
(i)].
In
particular,
|log(q)|
>
0
is
easily
computed
in
terms
of
the
various
q-parameters
of
the
elliptic
curve
E
F
[cf.
[IUTchI],
Definition
3.1,
(b)]
at
v
∈
V
bad
(
=
∅).
Then
it
holds
that
−
|log(Θ)|
∈
R,
and
−
|log(Θ)|
≥
−
|log(q)|
—
i.e.,
C
Θ
≥
−1
for
any
real
number
C
Θ
∈
R
such
that
−
|log(Θ)|
≤
C
Θ
·
|log(q)|.
Acknowledgements:
The
research
discussed
in
the
present
paper
profited
enormously
from
the
gen-
erous
support
that
the
author
received
from
the
Research
Institute
for
Mathematical
Sciences,
a
Joint
Usage/Research
Center
located
in
Kyoto
University.
At
a
personal
level,
I
would
like
to
thank
Fumiharu
Kato,
Akio
Tamagawa,
Go
Yamashita,
Mo-
hamed
Saı̈di,
Yuichiro
Hoshi,
Ivan
Fesenko,
Fucheng
Tan,
Emmanuel
Lepage,
Arata
Minamide,
and
Wojciech
Porowski
for
many
stimulating
discussions
concerning
the
material
presented
in
this
paper.
Also,
I
feel
deeply
indebted
to
Go
Yamashita,
Mohamed
Saı̈di,
and
Yuichiro
Hoshi
for
their
meticulous
reading
of
and
numerous
comments
concerning
the
present
paper.
Finally,
I
would
like
to
express
my
deep
gratitude
to
Ivan
Fesenko
for
his
quite
substantial
efforts
to
disseminate
—
for
in-
stance,
in
the
form
of
a
survey
that
he
wrote
—
the
theory
discussed
in
the
present
series
of
papers.
Notations
and
Conventions:
We
shall
continue
to
use
the
“Notations
and
Conventions”
of
[IUTchI],
§0.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
23
Section
1:
The
Log-theta-lattice
In
the
present
§1,
we
discuss
various
enhancements
to
the
theory
of
log-shells,
as
developed
in
[AbsTopIII].
In
particular,
we
develop
the
theory
of
the
log-link
[cf.
Definition
1.1;
Propositions
1.2,
1.3],
which,
together
with
the
Θ
×μ
-
and
Θ
×μ
gau
-links
of
[IUTchII],
Corollary
4.10,
(iii),
leads
naturally
to
the
construction
of
the
log-
theta-lattice,
an
apparatus
that
is
central
to
the
theory
of
the
present
series
of
papers.
We
conclude
the
present
§1
with
a
discussion
of
various
coric
structures
associated
to
the
log-theta-lattice
[cf.
Theorem
1.5].
In
the
following
discussion,
we
assume
that
we
have
been
given
initial
Θ-data
as
in
[IUTchI],
Definition
3.1.
We
begin
by
reviewing
various
aspects
of
the
theory
of
log-shells
developed
in
[AbsTopIII].
Definition
1.1.
Let
†
F
=
{
†
F
v
}
v∈V
be
an
F-prime-strip
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
Definition
5.2,
(i)].
Write
†
F
=
{
†
F
v
}
v∈V
;
†
×μ
F
=
{
†
F
v
×μ
}
v∈V
;
†
D
=
{
†
D
v
}
v∈V
for
the
associated
F
-,
F
×μ
-,
D-prime-strips
[cf.
[IUTchI],
Remark
5.2.1,
(ii);
[IUTchII],
Definition
4.9,
(vi),
(vii);
[IUTchI],
Remark
5.2.1,
(i)].
Recall
the
func-
torial
algorithm
of
[IUTchII],
Corollary
4.6,
(i),
in
the
F-prime-strip
†
F
for
con-
structing
the
assignment
Ψ
cns
(
†
F)
given
by
def
V
non
v
→
Ψ
cns
(
†
F)
v
=
G
v
(
†
Π
v
)
Ψ
†
F
v
V
arc
v
→
Ψ
cns
(
†
F)
v
=
Ψ
†
F
v
def
—
where
the
data
in
brackets
“{−}”
is
to
be
regarded
as
being
well-defined
only
up
to
a
†
Π
v
-conjugacy
indeterminacy
[cf.
[IUTchII],
Corollary
4.6,
(i),
for
more
details].
In
the
following,
we
shall
write
def
(−)
gp
=
(−)
gp
{0}
for
the
formal
union
with
{0}
of
the
groupification
(−)
gp
of
a
[multiplicatively
written]
monoid
“(−)”.
Thus,
by
setting
the
product
of
all
elements
of
(−)
gp
with
0
to
be
equal
to
0,
one
obtains
a
natural
monoid
structure
on
(−)
gp
.
(i)
Let
v
∈
V
non
.
Write
(Ψ
†
F
v
⊇
Ψ
×
→)
†
F
v
pf
Ψ
∼
=
(Ψ
×
†
F
†
F
)
v
v
def
pf
for
the
perfection
(Ψ
×
of
the
submonoid
of
units
Ψ
×
of
Ψ
†
F
v
.
Now
let
us
†
F
)
†
F
v
v
recall
from
the
theory
of
[AbsTopIII]
[cf.
[AbsTopIII],
Definition
3.1,
(iv);
[Ab-
sTopIII],
Proposition
3.2,
(iii),
(v)]
that
the
natural,
algorithmically
constructible
24
SHINICHI
MOCHIZUKI
gp
ind-topological
field
structure
on
Ψ
†
F
v
allows
one
to
define
a
p
v
-adic
logarithm
on
†
F
v
for
con-
Ψ
∼
†
F
,
which,
in
turn,
yields
a
functorial
algorithm
in
the
Frobenioid
v
∼
structing
an
ind-topological
field
structure
on
Ψ
†
F
v
.
Write
Ψ
log(
†
F
v
)
⊆
Ψ
∼
†
F
v
for
the
resulting
multiplicative
monoid
of
nonzero
integers.
Here,
we
observe
that
the
resulting
diagram
gp
Ψ
†
F
v
⊇
Ψ
×
→
Ψ
∼
=
Ψ
log(
†
F
v
)
†
F
†
F
v
v
is
compatible
with
the
various
natural
actions
of
†
Π
v
G
v
(
†
Π
v
)
on
each
of
the
[four]
“Ψ’s”
appearing
in
this
diagram.
The
pair
{
†
Π
v
Ψ
log(
†
F
v
)
}
now
determines
a
Frobenioid
log(
†
F
v
)
[cf.
[AbsTopIII],
Remark
3.1.1;
[IUTchI],
Remark
3.3.2]
—
which
is,
in
fact,
nat-
urally
isomorphic
to
the
Frobenioid
†
F
v
,
but
which
we
wish
to
think
of
as
being
related
to
†
F
v
via
the
above
diagram.
We
shall
denote
this
diagram
by
means
of
the
notation
log
†
F
v
−→
log(
†
F
v
)
and
refer
to
this
relationship
between
†
F
v
and
log(
†
F
v
)
as
the
tautological
log-
∼
link
associated
to
†
F
v
[or,
when
†
F
is
fixed,
at
v].
If
log(
†
F
v
)
→
‡
F
v
is
any
[poly-]isomorphism
of
Frobenioids,
then
we
shall
write
†
F
v
log
−→
‡
F
v
for
the
diagram
obtained
by
post-composing
the
tautological
log-link
associated
∼
to
†
F
v
with
the
given
[poly-]isomorphism
log(
†
F
v
)
→
‡
F
v
and
refer
to
this
re-
lationship
between
†
F
v
and
‡
F
v
as
a
log-link
from
†
F
v
to
‡
F
v
;
when
the
given
∼
[poly-]isomorphism
log(
†
F
v
)
→
‡
F
v
is
the
full
poly-isomorphism,
then
we
shall
re-
fer
to
the
resulting
log-link
as
the
full
log-link
from
†
F
v
to
‡
F
v
.
Finally,
we
recall
of
the
submonoid
from
[AbsTopIII],
Definition
3.1,
(iv),
that
the
image
in
Ψ
∼
†
F
v
constitutes
a
compact
topological
module,
which
we
of
G
v
(
†
Π
v
)-invariants
of
Ψ
×
†
F
v
def
def
shall
refer
to
as
the
pre-log-shell.
Write
p
∗
v
=
p
v
when
p
v
is
odd
and
p
∗
v
=
p
2
v
when
p
v
is
even.
Then
we
shall
refer
to
the
result
of
multiplying
the
pre-log-shell
by
the
factor
(p
∗
v
)
−1
as
the
log-shell
gp
I
†
F
v
⊆
Ψ
∼
=
Ψ
log(
†
F
v
)
†
F
v
[cf.
[AbsTopIII],
Definition
5.4,
(iii)].
In
particular,
by
applying
the
natural,
al-
gp
gorithmically
constructible
ind-topological
field
structure
on
Ψ
log(
†
F
v
)
[cf.
[Ab-
sTopIII],
Proposition
3.2,
(iii)],
it
thus
follows
that
one
may
think
of
this
log-shell
as
an
object
associated
to
the
codomain
of
any
[that
is
to
say,
not
necessarily
tautological!]
log-link
†
F
v
log
−→
‡
F
v
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
25
—
i.e.,
an
object
that
is
determined
by
the
image
of
a
certain
portion
[namely,
the
G
v
(
†
Π
v
)-invariants
of
Ψ
×
†
F
]
of
the
domain
of
this
log-link.
v
⊆
Ψ
gp
(ii)
Let
v
∈
V
arc
.
For
N
∈
N
≥1
,
write
Ψ
†
μ
F
N
v
⊆
Ψ
×
†
F
†
F
for
the
subgroup
v
v
of
N
-th
roots
of
unity
and
Ψ
∼
Ψ
gp
for
the
[pointed]
universal
covering
of
the
†
F
†
F
v
v
topological
group
determined
by
the
groupification
Ψ
gp
of
the
topological
monoid
†
F
v
Ψ
†
F
v
.
Then
one
verifies
immediately
that
one
may
think
of
the
composite
covering
of
topological
groups
μ
N
Ψ
gp
Ψ
gp
Ψ
∼
†
F
†
F
†
F
/Ψ
†
F
v
v
v
v
—
where
the
second
“”
is
the
natural
surjection
—
as
a
[pointed]
universal
cov-
μ
N
∼
as
an
object
constructed
ering
of
Ψ
gp
†
F
/Ψ
†
F
.
That
is
to
say,
one
may
think
of
Ψ
†
F
v
v
v
μ
N
[cf.
also
Remark
1.2.1,
(i),
below].
Now
let
us
recall
from
the
from
Ψ
gp
†
F
/Ψ
†
F
v
v
theory
of
[AbsTopIII]
[cf.
[AbsTopIII],
Definition
4.1,
(iv);
[AbsTopIII],
Propo-
sition
4.2,
(i),
(ii)]
that
the
natural,
algorithmically
constructible
[i.e.,
starting
from
the
collection
of
data
†
F
v
—
cf.
[IUTchI],
Definition
5.2,
(i),
(b)]
topological
gp
field
structure
on
Ψ
†
F
v
allows
one
to
define
a
[complex
archimedean]
logarithm
on
†
F
v
Ψ
∼
†
F
,
which,
in
turn,
yields
a
functorial
algorithm
in
the
collection
of
data
v
[cf.
[IUTchI],
Definition
5.2,
(i),
(b)]
for
constructing
a
topological
field
structure
def
∼
†
U
v
=
†
D
v
[cf.
[AbsTopIII],
on
Ψ
∼
†
F
,
together
with
a
Ψ
†
F
-Kummer
structure
on
v
v
Definition
4.1,
(iv);
[IUTchII],
Proposition
4.4,
(i)].
Write
Ψ
log(
†
F
v
)
⊆
Ψ
∼
†
F
v
for
the
resulting
multiplicative
monoid
of
nonzero
integral
elements
[i.e.,
elements
of
norm
≤
1].
Here,
we
observe
that
the
resulting
diagram
gp
Ψ
∼
=
Ψ
log(
†
F
v
)
Ψ
†
F
v
⊆
Ψ
gp
†
F
†
F
v
v
is
compatible
[cf.
the
discussion
of
[AbsTopIII],
Definition
4.1,
(iv)]
with
the
gp
co-holomorphicizations
determined
by
the
natural
Ψ
†
F
v
-Kummer
[cf.
[IUTchII],
the
above
discussion]
structures
on
Proposition
4.4,
(i)]
and
Ψ
∼
†
F
-Kummer
[cf.
v
†
U
v
.
The
triple
of
data
consisting
of
the
topological
monoid
Ψ
log(
†
F
v
)
,
the
Aut-
†
U
v
discussed
above
holomorphic
space
†
U
v
,
and
the
Ψ
∼
†
F
-Kummer
structure
on
v
determines
a
collection
of
data
[i.e.,
as
in
[IUTchI],
Definition
5.2,
(i),
(b)]
log(
†
F
v
)
which
is,
in
fact,
naturally
isomorphic
to
the
collection
of
data
†
F
v
,
but
which
we
wish
to
think
of
as
being
related
to
†
F
v
via
the
above
diagram.
We
shall
denote
this
diagram
by
means
of
the
notation
†
F
v
−→
log(
†
F
v
)
log
and
refer
to
this
relationship
between
†
F
v
and
log(
†
F
v
)
as
the
tautological
log-
∼
link
associated
to
†
F
v
[or,
when
†
F
is
fixed,
at
v].
If
log(
†
F
v
)
→
‡
F
v
is
any
26
SHINICHI
MOCHIZUKI
[poly-]isomorphism
of
collections
of
data
[i.e.,
as
in
[IUTchI],
Definition
5.2,
(i),
(b)],
then
we
shall
write
†
F
v
log
−→
‡
F
v
for
the
diagram
obtained
by
post-composing
the
tautological
log-link
associated
∼
to
†
F
v
with
the
given
[poly-]isomorphism
log(
†
F
v
)
→
‡
F
v
and
refer
to
this
re-
lationship
between
†
F
v
and
‡
F
v
as
a
log-link
from
†
F
v
to
‡
F
v
;
when
the
given
∼
[poly-]isomorphism
log(
†
F
v
)
→
‡
F
v
is
the
full
poly-isomorphism,
then
we
shall
re-
fer
to
the
resulting
log-link
as
the
full
log-link
from
†
F
v
to
‡
F
v
.
Finally,
we
recall
from
[AbsTopIII],
Definition
4.1,
(iv),
that
the
submonoid
of
units
Ψ
×
⊆
Ψ
†
F
v
†
F
v
determines
a
compact
topological
subquotient
of
Ψ
∼
†
F
,
which
we
shall
refer
to
as
v
-orbit
of
the
[uniquely
determined]
the
pre-log-shell.
We
shall
refer
to
the
Ψ
×
log(
†
F
v
)
which
is
preserved
by
multiplication
by
±1
and
whose
closed
line
segment
of
Ψ
∼
†
F
v
endpoints
differ
by
a
generator
of
the
kernel
of
the
natural
surjection
Ψ
∼
Ψ
gp
†
F
†
F
v
v
—
or,
equivalently,
the
Ψ
×
-orbit
of
the
result
of
multiplying
by
N
the
[uniquely
log(
†
F
v
)
which
is
preserved
by
multiplication
by
±1
determined]
closed
line
segment
of
Ψ
∼
†
F
v
and
whose
endpoints
differ
by
a
generator
of
the
kernel
of
the
natural
surjection
μ
N
Ψ
gp
—
as
the
log-shell
Ψ
∼
†
F
†
F
/Ψ
†
F
v
v
v
gp
I
†
F
v
⊆
Ψ
∼
=
Ψ
log(
†
F
v
)
†
F
v
[cf.
[AbsTopIII],
Definition
5.4,
(v)].
Thus,
one
may
think
of
the
log-shell
as
an
μ
N
object
constructed
from
Ψ
gp
†
F
/Ψ
†
F
.
Moreover,
by
applying
the
natural,
algorithmi-
v
v
gp
cally
constructible
topological
field
structure
on
Ψ
log(
†
F
v
)
(=
Ψ
∼
†
F
),
it
thus
follows
v
that
one
may
think
of
this
log-shell
as
an
object
associated
to
the
codomain
of
any
[that
is
to
say,
not
necessarily
tautological!]
log-link
†
F
v
log
−→
‡
F
v
—
i.e.,
an
object
that
is
determined
by
the
image
of
a
certain
portion
[namely,
the
subquotient
Ψ
×
of
Ψ
∼
†
F
]
of
the
domain
of
this
log-link.
†
F
v
v
(iii)
Write
†
def
log(
F)
=
†
def
log(
F
v
)
=
Ψ
∼
†
F
v
v∈V
for
the
collection
of
ind-topological
modules
constructed
in
(i),
(ii)
above
indexed
by
v
∈
V
—
where
the
group
structure
arises
from
the
additive
portion
of
the
field
discussed
in
(i),
(ii);
for
v
∈
V
non
,
we
regard
Ψ
∼
as
equipped
structures
on
Ψ
∼
†
F
†
F
v
v
with
its
natural
G
v
(
†
Π
v
)-action.
Write
log(
†
F)
=
{log(
†
F
v
)}
v∈V
def
for
the
F-prime-strip
determined
by
the
data
log(
†
F
v
)
constructed
in
(i),
(ii)
for
v
∈
V.
We
shall
denote
by
log
†
F
−→
log(
†
F)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
27
log
the
collection
of
diagrams
{
†
F
v
−→
log(
†
F
v
)}
v∈V
constructed
in
(i),
(ii)
for
v
∈
V
and
refer
to
this
relationship
between
†
F
and
log(
†
F)
as
the
tautological
log-link
∼
associated
to
†
F.
If
log(
†
F)
→
‡
F
is
any
[poly-]isomorphism
of
F-prime-strips,
then
we
shall
write
log
†
F
−→
‡
F
for
the
diagram
obtained
by
post-composing
the
tautological
log-link
associated
to
∼
F
with
the
given
[poly-]isomorphism
log(
†
F)
→
‡
F
and
refer
to
this
relationship
between
†
F
and
‡
F
as
a
log-link
from
†
F
to
‡
F;
when
the
given
[poly-]isomorphism
∼
log(
†
F)
→
‡
F
is
the
full
poly-isomorphism,
then
we
shall
refer
to
the
resulting
log-
link
as
the
full
log-link
from
†
F
to
‡
F.
Finally,
we
shall
write
†
def
I
†
F
=
{I
†
F
v
}
v∈V
for
the
collection
of
log-shells
constructed
in
(i),
(ii)
for
v
∈
V
and
refer
to
this
collection
as
the
log-shell
associated
to
†
F
and
[by
a
slight
abuse
of
notation]
I
†
F
⊆
log(
†
F)
for
the
collection
of
natural
inclusions
indexed
by
v
∈
V.
In
particular,
[cf.
the
discussion
of
(i),
(ii)],
it
thus
follows
that
one
may
think
of
this
log-shell
as
an
object
associated
to
the
codomain
of
any
[that
is
to
say,
not
necessarily
tautological!]
log-
link
log
†
F
−→
‡
F
—
i.e.,
an
object
that
is
determined
by
the
image
of
a
certain
portion
[cf.
the
discussion
of
(i),
(ii)]
of
the
domain
of
this
log-link.
(iv)
Let
v
∈
V
non
.
Then
observe
that
it
follows
immediately
from
the
construc-
tions
of
(i)
that
the
ind-topological
modules
with
G
v
(
†
Π
v
)-action
I
†
F
v
⊆
log(
†
F
v
)
may
be
constructed
solely
from
the
collection
of
data
†
F
v
×μ
[i.e.,
the
portion
of
the
F
×μ
-prime-strip
†
F
×μ
labeled
by
v].
That
is
to
say,
in
light
of
the
definition
of
a
×μ-Kummer
structure
[cf.
[IUTchII],
Definition
4.9,
(i),
(ii),
(iv),
(vi),
(vii)],
these
constructions
only
require
the
perfection
“(−)
pf
”
of
the
units
and
are
manifestly
unaffected
by
the
operation
of
forming
the
quotient
by
a
torsion
subgroup
of
the
units.
Write
I
†
F
v
×μ
⊆
log(
†
F
v
×μ
)
for
the
resulting
ind-topological
modules
with
G
v
(
†
Π
v
)-action,
regarded
as
objects
constructed
from
†
F
v
×μ
.
(v)
Let
v
∈
V
arc
.
Then
by
applying
the
algorithms
for
constructing
“k
∼
(G)”,
“I(G)”
given
in
[AbsTopIII],
Proposition
5.8,
(v),
to
the
[object
of
the
category
“TM
”
of
split
topological
monoids
discussed
in
[IUTchI],
Example
3.4,
(ii),
deter-
mined
by
the]
split
Frobenioid
portion
of
the
collection
of
data
†
F
v
,
one
obtains
a
functorial
algorithm
in
the
collection
of
data
†
F
v
for
constructing
a
topological
module
log(
†
F
v
)
[i.e.,
corresponding
to
“k
∼
(G)”]
and
a
topological
subspace
I
†
F
v
[i.e.,
corresponding
to
“I(G)”].
In
fact,
this
functorial
algorithm
only
makes
use
of
the
unit
portion
of
this
split
Frobenioid,
together
with
a
pointed
universal
covering
28
SHINICHI
MOCHIZUKI
of
this
unit
portion.
Moreover,
by
arguing
as
in
(ii),
one
may
in
fact
regard
this
functorial
algorithm
as
an
algorithm
that
only
makes
use
of
the
quotient
of
this
unit
portion
by
its
N
-torsion
subgroup,
for
N
∈
N
≥1
,
together
with
a
pointed
universal
covering
of
this
quotient.
That
is
to
say,
this
functorial
algorithm
may,
in
fact,
be
regarded
as
a
functorial
algorithm
in
the
collection
of
data
†
F
v
×μ
[cf.
Remark
1.2.1,
(i),
below;
[IUTchII],
Definition
4.9,
(v),
(vi),
(vii)].
Write
⊆
I
†
F
v
×μ
log(
†
F
v
×μ
)
for
the
resulting
topological
module
equipped
with
a
closed
subspace,
regarded
as
objects
constructed
from
†
F
v
×μ
.
(vi)
Finally,
just
as
in
(iii),
we
shall
write
I
†
F
×μ
def
=
{I
†
F
v
×μ
}
v∈V
⊆
log(
†
F
×μ
)
def
=
{log(
†
F
v
×μ
)}
v∈V
for
the
resulting
collections
of
data
constructed
solely
from
the
F
×μ
-prime-strip
†
×μ
F
[i.e.,
which
we
do
not
regard
as
objects
constructed
from
†
F!].
Remark
1.1.1.
(i)
Thus,
log-links
may
be
thought
of
as
correspondences
between
certain
portions
of
the
ind-topological
monoids
in
the
domain
of
the
log-link
and
certain
portions
of
the
ind-topological
monoids
in
the
codomain
of
the
log-link.
Frequently,
in
the
theory
of
the
present
paper,
we
shall
have
occasion
to
consider
“iterates”
of
log-links.
The
log-links
—
i.e.,
correspondences
between
certain
portions
of
the
ind-
topological
monoids
in
the
domains
and
codomains
of
the
log-links
—
that
appear
in
such
iterates
are
to
be
understood
as
being
defined
only
on
the
[local]
units
[cf.
also
(ii)
below,
in
the
case
of
v
∈
V
arc
]
that
appear
in
the
domains
of
these
log-links.
Thus,
for
instance,
when
considering
[the
nonzero
elements
of]
a
global
number
field
embedded
in
an
“idèlic”
product
[indexed
by
the
set
of
all
valuations
of
the
number
field]
of
localizations,
we
shall
regard
the
log-links
that
appear
as
being
defined
only
on
the
product
[indexed
by
the
set
of
all
valuations
of
the
number
field]
of
the
groups
of
local
units
that
appear
in
the
domains
of
these
log-links.
Indeed,
in
the
theory
of
the
present
paper,
the
only
reason
for
the
introduction
of
log-links
is
to
render
possible
the
construction
of
the
log-shells
from
the
various
[local]
units.
That
is
to
say,
the
construction
of
log-shells
does
not
require
the
use
of
the
“non-
unit”
—
i.e.,
the
local
and
global
“value
group”
—
portions
of
the
various
monoids
in
the
domain.
Thus,
when
considering
the
effect
of
applying
various
iterates
of
log-
links,
it
suffices,
from
the
point
of
view
of
computing
the
effect
of
the
construction
of
the
log-shells
from
the
local
units,
to
consider
the
effect
of
such
iterates
on
the
various
groups
of
local
units
that
appear.
(ii)
Suppose
that
we
are
in
the
situation
of
the
discussion
of
[local]
units
in
(i),
in
the
case
of
v
∈
V
arc
.
Then,
when
thinking
of
Kummer
structures
in
terms
of
Aut-holomorphic
structures
and
co-holomorphicizations,
as
in
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
29
discussion
of
[IUTchI],
Remark
3.4.2,
it
is
natural
to
regard
the
[local]
units
that
appear
as
being,
in
fact,
“Aut-holomorphic
semi-germs”,
that
is
to
say,
·
projective
systems
of
arbitrarily
small
neighborhoods
of
the
[local]
units
[i.e.,
of
“S
1
”
in
“C
×
”,
or,
in
the
notation
of
[IUTchI],
Example
3.4,
(i);
[IUTchI],
Remark
3.4.2,
of
“O
×
(C
v
)”
in
“O
(C
v
)
gp
”],
equipped
with
·
the
Aut-holomorphic
structures
induced
by
restricting
the
ambient
Aut-
holomorphic
structure
[i.e.,
of
“C
×
”,
or,
in
the
notation
of
[IUTchI],
Ex-
ample
3.4,
(i);
[IUTchI],
Remark
3.4.2,
of
“O
(C
v
)
gp
”],
·
the
group
structure
[germ]
induced
by
restricting
the
ambient
group
structure
[i.e.,
of
“C
×
”,
or,
in
the
notation
of
[IUTchI],
Example
3.4,
(i);
[IUTchI],
Remark
3.4.2,
of
“O
(C
v
)
gp
”],
and
·
a
choice
of
one
of
the
two
connected
components
of
the
complement
of
the
units
in
a
sufficiently
small
neighborhood
[i.e.,
determined
by
“O
C
\
S
1
⊆
C
×
\
S
1
”,
or,
in
the
notation
of
[IUTchI],
Example
3.4,
(i);
[IUTchI],
Remark
3.4.2,
by
“O
(C
v
)
\
O
×
(C
v
)
⊆
O
(C
v
)
gp
\
O
×
(C
v
)”].
Indeed,
one
verifies
immediately
that
such
“Aut-holomorphic
semi-germs”
are
rigid
in
the
sense
that
they
do
not
admit
any
nontrivial
holomorphic
automorphisms.
In
particular,
by
thinking
of
the
[local]
units
as
“Aut-holomorphic
semi-germs”
in
this
way,
the
approach
to
Kummer
structures
in
terms
of
Aut-holomorphic
structures
and
co-holomorphicizations
discussed
in
[IUTchI],
Remark
3.4.2,
carries
over
without
change
[cf.
[AbsTopIII],
Corollary
2.3,
(i)].
Moreover,
in
light
of
the
well-known
discreteness
of
the
image
of
the
units
of
a
number
field
via
the
logarithms
of
the
various
archimedean
valuations
of
the
number
field
[cf.,
e.g.,
[Lang],
p.
144,
Theorem
5],
it
follows
that
the
“idèlic”
aspects
discussed
in
(i)
are
also
unaffected
by
thinking
in
terms
of
Aut-holomorphic
semi-germs.
Remark
1.1.2.
(i)
In
the
notation
of
Definition
1.1,
(i),
we
observe
that
the
tautological
log-link
log
†
F
v
−→
log(
†
F
v
)
satisfies
the
property
that
there
is
a
tautological
identification
between
the
†
Π
v
’s
that
appear
in
the
data
that
gives
rise
to
the
domain
[i.e.,
{
†
Π
v
Ψ
†
F
v
}]
and
the
data
that
gives
rise
to
the
codomain
[i.e.,
{
†
Π
v
Ψ
log(
†
F
v
)
}]
of
the
tautological
log-link.
By
contrast,
the
†
Π
v
that
appears
in
the
data
that
gives
rise
to
the
domain
of
the
full
log-link
log
†
F
v
−→
‡
F
v
is
related
to
the
‡
Π
v
[where
we
use
analogous
notational
conventions
for
objects
associated
to
‡
F
to
the
notational
conventions
already
in
force
for
objects
associated
to
†
F]
that
appears
in
the
data
that
gives
rise
to
the
codomain
of
the
full
log-link
∼
by
means
of
a
full
poly-isomorphism
†
Π
v
→
‡
Π
v
.
In
this
situation,
the
specific
isomorphism
between
the
†
Π
v
’s
in
the
domain
and
codomain
of
the
tautological
log-link
may
be
thought
of
as
a
sort
of
specific
“rigid-
ifying
path”
between
the
†
Π
v
’s
in
the
domain
and
codomain
of
the
tau-
tological
log-link
that
is
constructed
precisely
by
using
[in
an
essential
30
SHINICHI
MOCHIZUKI
way!]
Frobenius-like
monoids
that
are
related
via
the
p
v
-adic
logarithm
[cf.
the
construction
of
Definition
1.1,
(i)],
i.e.,
by
applying
the
Galois-
equivariance
of
the
power
series
defining
the
p
v
-adic
logarithm
to
relate
automorphisms
of
the
monoid
Ψ
†
F
v
to
[induced!]
automorphisms
of
the
gp
=
Ψ
log(
†
F
v
)
.
monoid
Ψ
∼
†
F
v
Here,
the
use
of
the
term
“path”
is
intended
to
be
in
the
spirit
of
the
notion
of
a
path
in
the
étale
groupoid
[i.e.,
in
the
context
of
the
classical
theory
of
the
étale
fundamental
group],
except
that,
in
the
present
context,
we
allow
arbitrary
au-
tomorphism
indeterminacies,
instead
of
just
inner
automorphism
indeterminacies.
In
the
present
paper,
we
shall
work
mainly
with
the
full
log-link
[i.e.,
not
with
the
tautological
log-link!]
since,
in
the
context
of
the
multiradial
algorithms
to
be
developed
in
§3
below,
it
will
be
of
crucial
importance
to
be
able
to
express
the
relationship
between
the
étale-like
(−)
Π
v
’s
in
the
domain
and
codomain
of
the
log-links
that
appear
in
purely
étale-like
terms,
i.e.,
in
a
fashion
that
is
[unlike
the
specific
“rigidifying
path”
discussed
above!]
free
of
any
dependence
on
the
Frobenius-like
monoids
involved.
This
is
precisely
what
is
achieved
by
working
with
the
“tautologically
indeterminate
isomorphism”
between
(−)
Π
v
’s
that
underlies
the
full
log-link.
(ii)
An
analogous
discussion
to
that
of
(i)
may
be
given
in
the
situation
of
Definition
1.1,
(ii),
i.e.,
in
the
case
of
v
∈
V
arc
.
We
leave
the
routine
details
to
the
reader.
From
the
point
of
view
of
the
present
series
of
papers,
the
theory
of
[AbsTopIII]
may
be
summarized
as
follows.
Proposition
1.2.
(log-links
Between
F-prime-strips)
Let
†
F
=
{
†
F
v
}
v∈V
;
‡
F
=
{
‡
F
v
}
v∈V
be
F-prime-strips
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
Definition
5.2,
(i)]
and
†
F
log
−→
‡
F
a
log-link
from
†
F
to
‡
F.
Write
†
F
×μ
,
‡
F
×μ
for
the
associated
F
×μ
-prime-strips
[cf.
[IUTchII],
Definition
4.9,
(vi),
(vii)];
†
D,
‡
D
for
the
associated
D-prime-strips
[cf.
[IUTchI],
Remark
5.2.1,
(i)];
†
D
,
‡
D
for
the
associated
D
-prime-strips
[cf.
[IUTchI],
Definition
4.1,
(iv)].
Also,
let
us
recall
the
diagrams
gp
∼
gp
∼
→
log(
†
F
v
)
=
Ψ
log(
†
F
v
)
→
Ψ
†
F
v
⊇
Ψ
×
†
F
v
Ψ
†
F
v
⊆
Ψ
gp
log(
†
F
v
)
=
Ψ
log(
†
F
v
)
→
†
F
v
gp
Ψ
‡
F
v
gp
Ψ
‡
F
v
(∗
non
)
(∗
arc
)
—
where
the
v
of
(∗
non
)
(respectively,
(∗
arc
))
belongs
to
V
non
(respectively,
V
arc
),
log
and
the
[poly-]isomorphisms
on
the
right
are
induced
by
the
“
−→
”
—
of
Definition
1.1,
(i),
(ii).
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
(i)
(Coricity
of
Associated
D-Prime-Strips)
The
log-link
†
F
induces
[poly-]isomorphisms
†
∼
‡
D
→
D;
†
∼
D
→
‡
31
log
‡
−→
F
D
between
the
associated
D-
and
D
-prime-strips.
In
particular,
the
[poly-]isomorphism
†
∼
D
→
‡
D
induced
by
†
F
log
−→
‡
F
induces
a
[poly-]isomorphism
∼
Ψ
cns
(
†
D)
→
Ψ
cns
(
‡
D)
between
the
collections
of
monoids
equipped
with
auxiliary
data
of
[IUTchII],
Corol-
lary
4.5,
(i).
(ii)
(Simultaneous
Compatibility
with
Ring
Structures)
At
v
∈
V
non
,
the
natural
†
Π
v
-actions
on
the
“Ψ’s”
appearing
in
the
diagram
(∗
non
)
are
compat-
gp
gp
ible
with
the
ind-topological
ring
structures
on
Ψ
†
F
v
and
Ψ
log(
†
F
v
)
.
At
v
∈
V
arc
,
gp
gp
the
co-holomorphicizations
determined
by
the
natural
Ψ
†
F
v
-
and
Ψ
log(
†
F
v
)
(=
†
Ψ
∼
†
F
)-Kummer
structures
on
U
v
—
which
[cf.
the
discussion
of
Definition
1.1,
v
(ii)]
are
compatible
with
the
diagram
(∗
arc
)
—
are
compatible
with
the
topological
gp
gp
ring
structures
on
Ψ
†
F
v
and
Ψ
log(
†
F
v
)
.
(iii)
(Simultaneous
Compatibility
with
Log-volumes)
At
v
∈
V
non
,
the
diagram
(∗
non
)
is
compatible
with
the
natural
p
v
-adic
log-volumes
[cf.
[Ab-
sTopIII],
Proposition
5.7,
(i),
(c);
[AbsTopIII],
Corollary
5.10,
(ii)]
on
the
sub-
gp
gp
sets
of
†
Π
v
-invariants
of
Ψ
†
F
v
and
Ψ
log(
†
F
v
)
.
At
v
∈
V
arc
,
the
diagram
(∗
arc
)
is
compatible
with
the
natural
angular
log-volume
[cf.
Remark
1.2.1,
(i),
below;
[AbsTopIII],
Proposition
5.7,
(ii);
[AbsTopIII],
Corollary
5.10,
(ii)]
on
Ψ
×
and
†
F
v
the
natural
radial
log-volume
[cf.
[AbsTopIII],
Proposition
5.7,
(ii),
(c);
[Ab-
gp
sTopIII],
Corollary
5.10,
(ii)]
on
Ψ
log(
†
F
v
)
—
cf.
also
Remark
1.2.1,
(ii),
below.
(iv)
(Kummer
theory)
The
Kummer
isomorphisms
Ψ
cns
(
†
F)
∼
→
Ψ
cns
(
†
D);
∼
Ψ
cns
(
‡
F)
→
Ψ
cns
(
‡
D)
of
[IUTchII],
Corollary
4.6,
(i),
fail
to
be
compatible
with
the
[poly-]isomorphism
∼
Ψ
cns
(
†
D)
→
Ψ
cns
(
‡
D)
of
(i),
relative
to
the
diagrams
(∗
non
),
(∗
arc
)
[and
the
notational
conventions
of
Definition
1.1]
—
cf.
[AbsTopIII],
Corollary
5.5,
(iv).
[Here,
we
regard
the
diagrams
(∗
non
),
(∗
arc
)
as
diagrams
that
relate
Ψ
†
F
v
and
Ψ
‡
F
v
,
∼
log
via
the
[poly-]isomorphism
log(
†
F)
→
‡
F
that
determines
the
log-link
†
F
−→
‡
F.]
(v)
(Holomorphic
Log-shells)
At
v
∈
V
non
,
the
log-shell
I
†
F
v
⊆
log(
†
F
v
)
∼
(
→
gp
Ψ
‡
F
v
)
satisfies
the
following
properties:
(a
non
)
I
†
F
v
is
compact,
hence
of
finite
log-
volume
[cf.
[AbsTopIII],
Corollary
5.10,
(i)];
(b
non
)
I
†
F
v
contains
the
submonoid
32
SHINICHI
MOCHIZUKI
of
†
Π
v
-invariants
of
Ψ
log(
†
F
v
)
[cf.
[AbsTopIII],
Definition
5.4,
(iii)];
(c
non
)
I
†
F
v
arc
contains
the
image
of
the
submonoid
of
†
Π
v
-invariants
of
Ψ
×
,
the
†
F
.
At
v
∈
V
v
log-shell
gp
∼
I
†
F
v
⊆
log(
†
F
v
)
(
→
Ψ
‡
F
v
)
satisfies
the
following
properties:
(a
arc
)
I
†
F
v
is
compact,
hence
of
finite
radial
log-volume
[cf.
[AbsTopIII],
Corollary
5.10,
(i)];
(b
arc
)
I
†
F
v
contains
Ψ
log(
†
F
v
)
[cf.
[AbsTopIII],
Definition
5.4,
(v)];
(c
arc
)
the
image
of
I
†
F
v
in
Ψ
gp
contains
†
F
v
Ψ
×
[i.e.,
in
essence,
the
pre-log-shell].
†
F
v
(vi)
(Nonarchimedean
Mono-analytic
Log-shells)
At
v
∈
V
non
,
if
we
write
†
D
v
=
B(
†
G
v
)
0
for
the
portion
of
†
D
indexed
by
v
[cf.
the
notation
of
[IUTchII],
Corollary
4.5],
then
the
algorithms
for
constructing
“k
∼
(G)”,
“I(G)”
given
in
[AbsTopIII],
Proposition
5.8,
(ii),
yield
a
functorial
algorithm
in
the
category
†
D
v
for
constructing
an
ind-topological
module
equipped
with
a
continuous
†
G
v
-action
def
log(
†
D
v
)
=
†
G
v
k
∼
(
†
G
v
)
and
a
topological
submodule
—
i.e.,
a
“mono-analytic
log-shell”
—
I
†
D
v
=
I(
†
G
v
)
⊆
k
∼
(
†
G
v
)
def
equipped
with
a
p
v
-adic
log-volume
[cf.
[AbsTopIII],
Corollary
5.10,
(iv)].
More-
over,
there
is
a
natural
functorial
algorithm
[cf.
the
second
display
of
[IUTchII],
Corollary
4.6,
(ii)]
in
the
collection
of
data
†
F
v
×μ
[i.e.,
the
portion
of
†
F
×μ
labeled
by
v]
for
constructing
an
Ism-orbit
of
isomorphisms
[cf.
[IUTchII],
Example
1.8,
(iv);
[IUTchII],
Definition
4.9,
(i),
(vii)]
∼
log(
†
D
v
)
→
log(
†
F
v
×μ
)
of
ind-topological
modules
[cf.
Definition
1.1,
(iv)],
as
well
as
a
functorial
al-
gorithm
[cf.
[AbsTopIII],
Corollary
5.10,
(iv),
(c),
(d);
the
fourth
display
of
[IUTchII],
Corollary
4.5,
(ii);
the
final
display
of
[IUTchII],
Corollary
4.6,
(i)]
in
the
collection
of
data
†
F
v
for
constructing
isomorphisms
∼
∼
log(
†
D
v
)
→
log(
†
F
v
×μ
)
→
log(
†
F
v
)
∼
(
→
gp
Ψ
‡
F
v
)
of
ind-topological
modules.
The
various
isomorphisms
of
the
last
two
displays
are
compatible
with
one
another,
as
well
as
with
the
respective
†
G
v
-
and
G
v
(
†
Π
v
)-
actions
[relative
to
the
natural
identification
†
G
v
=
G
v
(
†
Π
v
)
that
arises
from
re-
garding
†
D
v
as
an
object
constructed
from
†
D
v
],
the
respective
log-shells,
and
the
respective
log-volumes
on
these
log-shells.
(vii)
(Archimedean
Mono-analytic
Log-shells)
At
v
∈
V
arc
,
the
algo-
rithms
for
constructing
“k
∼
(G)”,
“I(G)”
given
in
[AbsTopIII],
Proposition
5.8,
(v),
yield
a
functorial
algorithm
in
†
D
v
[regarded
as
an
object
of
the
category
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
33
“TM
”
of
split
topological
monoids
discussed
in
[IUTchI],
Example
3.4,
(ii)]
for
constructing
a
topological
module
log(
†
D
v
)
=
k
∼
(
†
G
v
)
def
and
a
topological
subspace
—
i.e.,
a
“mono-analytic
log-shell”
—
I
†
D
v
=
I(
†
G
v
)
⊆
k
∼
(
†
G
v
)
def
equipped
with
angular
and
radial
log-volumes
[cf.
[AbsTopIII],
Corollary
5.10,
(iv)].
Moreover,
there
is
a
natural
functorial
algorithm
[cf.
the
second
display
of
[IUTchII],
Corollary
4.6,
(ii)]
in
the
collection
of
data
†
F
v
×μ
for
constructing
a
poly-isomorphism
[i.e.,
an
orbit
of
isomorphisms
with
respect
to
the
indepen-
dent
actions
of
{±1}
on
each
of
the
direct
factors
that
occur
in
the
construction
of
[AbsTopIII],
Proposition
5.8,
(v)]
∼
log(
†
D
v
)
→
log(
†
F
v
×μ
)
of
topological
modules
[cf.
Definition
1.1,
(v)],
as
well
as
a
functorial
algorithm
[cf.
[AbsTopIII],
Corollary
5.10,
(iv),
(c),
(d);
the
fourth
display
of
[IUTchII],
Corollary
4.5,
(ii);
the
final
display
of
[IUTchII],
Corollary
4.6,
(i)]
in
the
collection
of
data
†
F
v
for
constructing
poly-isomorphisms
[i.e.,
orbits
of
isomorphisms
with
respect
to
the
independent
actions
of
{±1}
on
each
of
the
direct
factors
that
occur
in
the
construction
of
[AbsTopIII],
Proposition
5.8,
(v)]
∼
∼
∼
log(
†
D
v
)
→
log(
†
F
v
×μ
)
→
log(
†
F
v
)
gp
(
→
Ψ
‡
F
v
)
of
topological
modules.
The
various
isomorphisms
of
the
last
two
displays
are
com-
patible
with
one
another,
as
well
as
with
the
respective
log-shells
and
the
respec-
tive
angular
and
radial
log-volumes
on
these
log-shells.
(viii)
(Mono-analytic
Log-shells)
The
various
[poly-]isomorphisms
of
(vi),
(vii)
[cf.
also
Definition
1.1,
(iii),
(vi)]
yield
collections
of
[poly-]isomorphisms
indexed
by
v
∈
V
∼
log(
†
D
)
=
{log(
†
D
v
)}
v∈V
→
log(
†
F
×μ
)
=
{log(
†
F
v
×μ
)}
v∈V
def
def
∼
def
def
I
†
D
=
{I
†
D
v
}
v∈V
→
I
†
F
×μ
=
{I
†
F
v
×μ
}
v∈V
∼
∼
log(
†
D
)
→
log(
†
F
×μ
)
→
log(
†
F)
=
{log(
†
F
v
)}
v∈V
∼
→
∼
∼
def
gp
gp
Ψ
cns
(
‡
F)
=
{Ψ
‡
F
v
}
v∈V
def
def
I
†
D
→
I
†
F
×μ
→
I
†
F
=
{I
†
F
v
}
v∈V
gp
gp
—
where,
in
the
definition
of
“Ψ
cns
(
‡
F)”,
we
regard
each
Ψ
‡
F
v
,
for
v
∈
V
non
,
as
being
equipped
with
its
natural
G
v
(
‡
Π
v
)-action
[cf.
the
discussion
at
the
beginning
of
Definition
1.1].
34
SHINICHI
MOCHIZUKI
(ix)
(Coric
Holomorphic
Log-shells)
Let
∗
D
be
a
D-prime-strip;
write
F(
∗
D)
for
the
F-prime-strip
naturally
determined
by
Ψ
cns
(
∗
D)
[cf.
[IUTchII],
Remark
4.5.1,
(i)].
Suppose
that
†
F
=
‡
F
=
F(
∗
D),
and
that
the
given
log-link
F(
∗
D)
=
†
log
F
−→
‡
F
=
F(
∗
D)
is
the
full
log-link.
Then
there
exists
a
functorial
algorithm
in
the
D-prime-strip
∗
D
for
constructing
a
collection
of
topological
subspaces
—
i.e.,
a
collection
of
“coric
holomorphic
log-shells”
—
def
I
∗
D
=
I
†
F
gp
gp
of
the
collection
Ψ
cns
(
∗
D),
which
may
be
naturally
identified
with
Ψ
cns
(
‡
F),
together
with
a
collection
of
natural
isomorphisms
[cf.
(viii);
the
fourth
display
of
[IUTchII],
Corollary
4.5,
(ii)]
∼
I
∗
D
→
I
∗
D
—
where
we
write
∗
D
for
the
D
-prime-strip
determined
by
∗
D.
(x)
(Frobenius-picture)
Let
{
n
F}
n∈Z
be
a
collection
of
distinct
F-prime-
strips
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
Definition
5.2,
(i)]
indexed
by
the
integers.
Write
{
n
D}
n∈Z
for
the
associated
D-prime-strips
[cf.
[IUTchI],
Remark
5.2.1,
(i)]
and
{
n
D
}
n∈Z
for
the
associated
D
-prime-strips
[cf.
[IUTchI],
Definition
4.1,
(iv)].
Then
the
full
log-links
n
F
n
∈
Z,
give
rise
to
an
infinite
chain
log
.
.
.
−→
(n−1)
log
F
−→
n
log
F
−→
(n+1)
log
−→
(n+1)
F,
for
log
F
−→
.
.
.
of
log-linked
F-prime-strips
which
induces
chains
of
full
poly-isomorphisms
∼
∼
∼
.
.
.
→
n
D
→
(n+1)
D
→
.
.
.
∼
∼
∼
.
.
.
→
n
D
→
(n+1)
D
→
.
.
.
and
on
the
associated
D-
and
D
-prime-strips
[cf.
(i)].
These
chains
may
be
represented
symbolically
as
an
oriented
graph
Γ
[cf.
[AbsTopIII],
§0]
...
→
...
•
→
•
↓
→
•
→
...
...
◦
log
—
i.e.,
where
the
horizontal
arrows
correspond
to
the
“
−→
’s”;
the
“•’s”
corre-
spond
to
the
“
n
F”;
the
“◦”
corresponds
to
the
“
n
D”,
identified
up
to
isomorphism;
the
vertical/diagonal
arrows
correspond
to
the
Kummer
isomorphisms
of
(iv).
This
oriented
graph
Γ
admits
a
natural
action
by
Z
[cf.
[AbsTopIII],
Corollary
5.5,
(v)]
—
i.e.,
a
translation
symmetry
—
that
fixes
the
“core”
◦,
but
it
does
not
admit
arbitrary
permutation
symmetries.
For
instance,
Γ
does
not
admit
an
automorphism
that
switches
two
adjacent
vertices,
but
leaves
the
remaining
vertices
fixed.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
35
Proof.
The
various
assertions
of
Proposition
1.2
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
1.2.1.
(i)
Suppose
that
we
are
in
the
situation
of
Definition
1.1,
(ii).
Then
at
the
level
of
metrics
—
i.e.,
which
give
rise
to
angular
log-volumes
as
in
Proposition
μ
N
is
equipped
with
the
metric
obtained
by
1.2,
(iii)
—
we
suppose
that
Ψ
gp
†
F
/Ψ
†
F
v
v
descending
the
metric
of
Ψ
gp
†
F
,
but
we
regard
the
object
v
μ
N
μ
N
Ψ
gp
[or
Ψ
×
†
F
/Ψ
†
F
†
F
/Ψ
†
F
]
as
being
equipped
with
a
“weight
N
”
v
v
v
v
μ
N
—
i.e.,
which
has
the
effect
of
ensuring
that
the
log-volume
of
Ψ
×
†
F
/Ψ
†
F
is
equal
v
v
to
that
of
Ψ
×
†
F
.
That
is
to
say,
this
convention
concerning
“weights”
ensures
that
v
μ
N
does
not
have
any
effect
on
various
computations
of
log-
working
with
Ψ
gp
†
F
/Ψ
†
F
v
v
volume.
(ii)
Although,
at
first
glance,
the
compatibility
with
archimedean
log-volumes
discussed
in
Proposition
1.2,
(iii),
appears
to
relate
“different
objects”
—
i.e.,
angu-
lar
versus
radial
log-volumes
—
in
the
domain
and
codomain
of
the
log-link
under
consideration,
in
fact,
this
compatibility
property
may
be
regarded
as
an
invari-
ance
property
—
i.e.,
that
relates
“similar
objects”
in
the
domain
and
codomain
of
the
log-link
under
consideration
—
by
reasoning
as
follows.
Let
k
be
a
complex
archimedean
field.
Write
O
k
×
⊆
k
for
the
group
of
elements
of
absolute
value
=
1
and
k
×
⊆
k
for
the
group
of
nonzero
elements.
In
the
following,
we
shall
use
the
term
“metric
on
k”
to
refer
to
a
Riemannian
metric
on
the
real
analytic
manifold
determined
by
k
that
is
compatible
with
the
two
natural
almost
complex
structures
on
this
real
analytic
manifold
and,
moreover,
is
invariant
with
respect
to
arbitrary
additive
translation
automorphisms
of
k.
In
passing,
we
note
that
any
metric
on
k
is
also
invariant
with
respect
to
multiplication
by
elements
∈
O
k
×
.
Next,
let
us
observe
that
the
metrics
on
k
naturally
form
a
torsor
over
R
>0
.
In
particular,
if
we
write
k
×
∼
=
O
k
×
×
R
>0
for
the
natural
direct
product
decomposition,
then
one
verifies
immediately
that
any
metric
on
k
is
uniquely
determined
either
by
its
restriction
to
O
k
×
⊆
k
or
by
its
restriction
to
R
>0
⊆
k.
Thus,
if
one
regards
the
compatibility
property
concerning
angular
and
radial
log-
volumes
discussed
in
Proposition
1.2,
(iii),
as
a
property
concerning
the
respective
restrictions
of
the
corresponding
uniquely
determined
metrics
[i.e.,
the
metrics
corresponding
to
the
respective
standard
norms
on
the
complex
archimedean
fields
under
consideration
—
cf.
[AbsTopIII],
Proposition
5.7,
(ii),
(a)],
then
this
compat-
ibility
property
discussed
in
Proposition
1.2,
(iii),
may
be
regarded
as
a
property
that
asserts
the
invariance
of
the
respective
natural
metrics
with
respect
to
the
“transformation”
constituted
by
the
log-link.
Remark
1.2.2.
Before
proceeding,
we
pause
to
consider
the
significance
of
the
various
properties
discussed
in
Proposition
1.2,
(v).
For
simplicity,
we
suppose
36
SHINICHI
MOCHIZUKI
that
“
†
F”
is
the
F-prime-strip
that
arises
from
the
data
constructed
in
[IUTchI],
Examples
3.2,
(iii);
3.3,
(i);
3.4,
(i)
[cf.
[IUTchI],
Definition
5.2,
(i)].
(i)
Suppose
that
v
∈
V
non
.
Thus,
K
v
[cf.
the
notation
of
[IUTchI],
Definition
def
3.1,
(e)]
is
a
mixed-characteristic
nonarchimedean
local
field.
Write
k
=
K
v
,
O
k
⊆
k
for
the
ring
of
integers
of
k,
O
k
×
⊆
O
k
for
the
group
of
units,
and
log
k
:
O
k
×
→
k
for
the
p
v
-adic
logarithm.
Then,
at
a
more
concrete
level
—
i.e.,
relative
to
the
notation
of
the
present
discussion
—
the
log-shell
“I
†
F
v
”
corresponds
to
the
submodule
def
I
k
=
(p
∗
v
)
−1
·
log
k
(O
k
×
)
⊆
k
—
where
p
∗
v
=
p
v
if
p
v
is
odd,
p
∗
v
=
p
2
v
if
p
v
is
even
—
while
the
properties
(b
non
),
(c
non
)
of
Proposition
1.2,
(v),
correspond,
respectively,
to
the
evident
inclusions
O
k
=
O
k
\
{0}
⊆
O
k
⊆
I
k
;
def
log
k
(O
k
×
)
⊆
I
k
of
subsets
of
k.
(ii)
Suppose
that
v
∈
V
arc
.
Thus,
K
v
[cf.
the
notation
of
[IUTchI],
Definition
def
3.1,
(e)]
is
a
complex
archimedean
field.
Write
k
=
K
v
,
O
k
⊆
k
for
the
subset
of
elements
of
absolute
value
≤
1,
O
k
×
⊆
O
k
for
the
group
of
elements
of
absolute
value
=
1,
and
exp
k
:
k
→
k
×
for
the
exponential
map.
Then,
at
a
more
concrete
level
—
i.e.,
relative
to
the
notation
of
the
present
discussion
—
the
log-shell
“I
†
F
v
”
corresponds
to
the
subset
def
I
k
=
{a
∈
k
|
|a|
≤
π}
⊆
k
of
elements
of
absolute
value
≤
π,
while
the
properties
(b
arc
),
(c
arc
)
of
Proposition
1.2,
(v),
correspond,
respectively,
to
the
evident
inclusions
O
k
=
O
k
\
{0}
⊆
O
k
⊆
I
k
;
def
O
k
×
⊆
exp
k
(I
k
)
—
where
we
note
the
slightly
different
roles
played,
in
the
archimedean
[cf.
the
present
(ii)]
and
nonarchimedean
[cf.
(i)]
cases,
by
the
exponential
and
logarithmic
functions,
respectively
[cf.
[AbsTopIII],
Remark
4.5.2].
(iii)
The
diagram
represented
by
the
oriented
graph
Γ
of
Proposition
1.2,
(x),
is,
of
course,
far
from
commutative
[cf.
Proposition
1.2,
(iv)]!
Ultimately,
however,
[cf.
the
discussion
of
Remark
1.4.1,
(ii),
below]
we
shall
be
interested
in
(a)
constructing
invariants
with
respect
to
the
Z-action
on
Γ
—
i.e.,
in
effect,
constructing
objects
via
functorial
algorithms
in
the
coric
D-prime-strips
“
n
D”
—
while,
at
the
same
time,
(b)
relating
the
corically
constructed
objects
of
(a)
to
the
non-coric
“
n
F”
via
the
various
Kummer
isomorphisms
of
Proposition
1.2,
(iv).
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
37
That
is
to
say,
from
the
point
of
view
of
(a),
(b),
the
content
of
the
inclusions
discussed
in
(i)
and
(ii)
above
may
be
interpreted,
at
v
∈
V
non
,
as
follows:
the
coric
holomorphic
log-shells
of
Proposition
1.2,
(ix),
contain
not
only
the
images,
via
the
Kummer
isomorphisms
[i.e.,
the
vertical/diagonal
arrows
of
Γ],
of
the
various
“O
”
at
v
∈
V
non
,
but
also
the
images,
via
the
composite
of
the
Kummer
isomorphisms
with
the
various
iterates
[cf.
Remark
1.1.1]
of
the
log-link
[i.e.,
the
horizontal
arrows
of
Γ],
of
the
portions
of
the
various
“O
”
at
v
∈
V
non
on
which
these
iterates
are
defined.
An
analogous
statement
in
the
case
of
v
∈
V
arc
may
be
formulated
by
adjusting
the
wording
appropriately
so
as
to
accommodate
the
latter
portion
of
this
statement,
which
corresponds
to
a
certain
surjectivity
—
we
leave
the
routine
details
to
the
reader.
Thus,
although
the
diagram
[corresponding
to]
Γ
fails
to
be
commutative,
the
coric
holomorphic
log-shells
involved
exhibit
a
sort
of
“upper
semi-
commutativity”
with
respect
to
containing/surjecting
onto
the
various
images
arising
from
composites
of
arrows
in
Γ.
(iv)
Note
that
although
the
diagram
Γ
admits
a
natural
“upper
semi-commu-
tativity”
interpretation
as
discussed
in
(iii)
above,
it
fails
to
admit
a
corresponding
“lower
semi-commutativity”
interpretation.
Indeed,
such
a
“lower
semi-commu-
tativity”
interpretation
would
amount
to
the
existence
of
some
sort
of
collection
of
portions
of
the
various
“O
’s”
involved
[cf.
the
discussion
of
(i),
(ii)
above]
—
i.e.,
a
sort
of
“core”
—
that
are
mapped
to
one
another
isomorphically
by
the
various
maps
“log
k
”/“exp
k
”
[cf.
the
discussion
of
(i),
(ii)
above]
in
a
fashion
that
is
compatible
with
the
various
Kummer
isomorphisms
that
appear
in
the
diagram
Γ.
On
the
other
hand,
it
is
difficult
to
see
how
to
construct
such
a
collection
of
portions
of
the
various
“O
’s”
involved.
(v)
Proposition
1.2,
(iii),
may
be
interpreted
in
the
spirit
of
the
discussion
of
(iii)
above.
That
is
to
say,
although
the
diagram
corresponding
to
Γ
fails
to
be
commutative,
it
is
nevertheless
“commutative
with
respect
to
log-volumes”,
in
the
sense
discussed
in
Proposition
1.2,
(iii).
This
“commutativity
with
respect
to
log-volumes”
allows
one
to
work
with
log-volumes
in
a
fashion
that
is
consistent
with
all
composites
of
the
various
arrows
of
Γ.
Log-volumes
will
play
an
important
role
in
the
theory
of
§3,
below,
as
a
sort
of
mono-analytic
version
of
the
notion
of
the
degree
of
a
global
arithmetic
line
bundle
[cf.
the
theory
of
[AbsTopIII],
§5].
(vi)
As
discussed
in
[AbsTopIII],
§I3,
the
log-links
of
Γ
may
be
thought
of
as
a
sort
of
“juggling
of
,
”
[i.e.,
of
the
two
combinatorial
dimensions
of
the
ring
structure
constituted
by
addition
and
multiplication].
The
“arithmetic
holomorphic
structure”
constituted
by
the
coric
D-prime-strips
is
immune
to
this
juggling,
and
hence
may
be
thought
as
representing
a
sort
of
quotient
of
the
horizontal
arrow
portion
of
Γ
by
the
action
of
Z
[cf.
(iii),
(a)]
—
i.e.,
at
the
level
of
abstract
oriented
graphs,
as
a
sort
of
“oriented
copy
of
S
1
”.
That
is
to
say,
the
horizontal
arrow
portion
of
Γ
may
be
thought
of
as
a
sort
of
“unraveling”
of
38
SHINICHI
MOCHIZUKI
this
“oriented
copy
of
S
1
”,
which
is
subject
to
the
“juggling
of
,
”
constituted
by
the
Z-action.
Here,
it
is
useful
to
recall
that
(a)
the
Frobenius-like
structures
constituted
by
the
monoids
that
appear
in
the
horizontal
arrow
portion
of
Γ
play
the
crucial
role
in
the
theory
of
the
present
series
of
papers
of
allowing
one
to
construct
such
“non-
ring/scheme-theoretic
filters”
as
the
Θ-link
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.2,
(ii)].
By
contrast,
(b)
the
étale-like
structures
constituted
by
the
coric
D-prime-strips
play
the
crucial
role
in
the
theory
of
the
present
series
of
papers
of
allowing
one
to
construct
objects
that
are
capable
of
“functorially
permeating”
such
non-ring/scheme-theoretic
filters
as
the
Θ-link
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.2,
(ii)].
Finally,
in
order
to
relate
the
theory
of
(a)
to
the
theory
of
(b),
one
must
avail
oneself
of
Kummer
theory
[cf.
(iii),
(b),
above].
mono-anabelian
coric
étale-like
structures
invariant
differential
dθ
on
S
1
post-anabelian
Frobenius-like
structures
coordinate
functions
dθ
on
Γ
•
Fig.
1.1:
Analogy
with
the
differential
geometry
of
S
1
(vii)
From
the
point
of
view
of
the
discussion
in
(vi)
above
of
the
“oriented
copy
of
S
1
”
obtained
by
forming
the
quotient
of
the
horizontal
arrow
portion
of
Γ
by
Z,
one
may
think
of
the
coric
étale-like
structures
of
Proposition
1.2,
(i)
—
as
well
as
the
various
objects
constructed
from
these
coric
étale-like
structures
via
the
various
mono-anabelian
algorithms
discussed
in
[AbsTopIII]
—
as
corresponding
to
the
“canonical
invariant
differential
dθ”
on
S
1
[which
is,
in
particular,
invariant
with
respect
to
the
action
of
Z!].
On
the
other
hand,
the
various
post-anabelian
Frobenius-like
structures
obtained
by
forgetting
the
mono-anabelian
algorithms
ap-
plied
to
construct
these
objects
—
cf.,
e.g..,
the
“Ψ
cns
(
†
F)”
that
appear
in
the
Kummer
isomorphisms
of
Proposition
1.2,
(iv)
—
may
be
thought
of
as
coordinate
functions
on
the
horizontal
arrow
portion
of
Γ
[which
are
not
invariant
with
respect
to
the
action
of
Z!]
of
the
form
“
•
dθ”
obtained
by
integrating
the
invariant
dif-
ferential
dθ
along
various
paths
of
Γ
that
emanate
from
some
fixed
vertex
“•”
of
Γ.
This
point
of
view
is
summarized
in
Fig.
1.1
above.
Finally,
we
observe
that
this
point
of
view
is
reminiscent
of
the
discussion
of
[AbsTopIII],
§I5,
concerning
the
analogy
between
the
theory
of
[AbsTopIII]
and
the
construction
of
canonical
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
39
coordinates
via
integration
of
Frobenius-invariant
differentials
in
the
classical
p-adic
theory.
Remark
1.2.3.
(i)
Observe
that,
relative
to
the
notation
of
Remark
1.2.2,
(i),
any
multi-
plicative
indeterminacy
with
respect
to
the
action
on
O
k
of
some
subgroup
H
⊆
O
k
×
at
some
“•”
of
the
diagram
Γ
gives
rise
to
an
additive
indeterminacy
with
respect
to
the
action
of
log
k
(H)
on
the
copy
of
“O
k
”
that
corresponds
to
the
subsequent
“•”
of
the
diagram
Γ.
In
particular,
if
H
consists
of
roots
of
unity,
then
log
k
(H)
=
{0},
so
the
resulting
additive
indeterminacy
ceases
to
exist.
This
observation
will
play
a
crucial
role
in
the
theory
of
§3,
below,
when
it
is
applied
in
the
context
of
the
constant
multiple
rigidity
properties
constituted
by
the
canon-
ical
splittings
of
theta
and
Gaussian
monoids
discussed
in
[IUTchII],
Proposition
3.3,
(i);
[IUTchII],
Corollary
3.6,
(iii)
[cf.
also
[IUTchII],
Corollary
1.12,
(ii);
the
discussion
of
[IUTchII],
Remark
1.12.2,
(iv)].
(ii)
In
the
theory
of
§3,
below,
we
shall
consider
global
arithmetic
line
bundles.
×
This
amounts,
in
effect,
to
considering
multiplicative
translates
by
f
∈
F
mod
of
the
product
of
the
various
“O
k
×
”
of
Remark
1.2.2,
(i),
(ii),
as
v
ranges
over
the
elements
of
V.
Such
translates
are
disjoint
from
one
another,
except
in
the
case
where
f
is
a
unit
at
all
v
∈
V.
By
elementary
algebraic
number
theory
[cf.,
e.g.,
[Lang],
p.
144,
the
proof
of
Theorem
5],
this
corresponds
precisely
to
the
case
where
f
is
a
root
of
unity.
In
particular,
to
consider
quotients
by
this
multiplicative
action
×
at
one
“•”
of
the
diagram
Γ
[where
we
allow
v
to
range
over
the
elements
by
F
mod
of
V]
gives
rise
to
an
additive
indeterminacy
by
“logarithms
of
roots
of
unity”
at
the
subsequent
“•”
of
the
diagram
Γ.
In
particular,
at
v
∈
V
non
,
the
resulting
additive
indeterminacy
ceases
to
exist
[cf.
the
discussion
of
(i);
Definition
1.1,
(iv)];
at
v
∈
V
arc
,
the
resulting
indeterminacy
corresponds
to
considering
certain
quotients
of
the
copies
of
“O
k
×
”
—
i.e.,
of
“S
1
”
—
that
appear
by
some
finite
subgroup
[cf.
the
discussion
of
Definition
1.1,
(ii)].
These
observations
will
be
of
use
in
the
development
of
the
theory
of
§3,
below.
Remark
1.2.4.
(i)
At
this
point,
we
pause
to
recall
the
important
observation
that
the
log-link
gp
gp
is
incompatible
with
the
ring
structures
of
Ψ
†
F
v
and
Ψ
log(
†
F
v
)
[cf.
the
notation
of
Proposition
1.2,
(ii)],
in
the
sense
that
it
does
not
arise
from
a
ring
homomorphism
between
these
two
rings.
The
barrier
constituted
by
this
incompatibility
between
the
ring
structures
on
either
side
of
the
log-link
is
precisely
what
is
referred
to
as
the
“log-wall”
in
the
theory
of
[AbsTopIII]
[cf.
the
discussion
of
[AbsTopIII],
§I4].
This
incompatibility
with
the
respective
ring
structures
implies
that
it
is
not
possible,
a
priori,
to
transport
objects
whose
structure
depends
on
these
ring
structures
via
the
log-link
by
invoking
the
principle
of
“transport
of
structure”.
From
the
point
of
view
of
the
theory
of
the
present
series
of
papers,
this
means,
in
particular,
that
the
log-wall
is
incompatible
with
conventional
scheme-theoretic
base-
points,
which
are
defined
by
means
of
geometric
points
[i.e.,
ring
homo-
morphisms
of
a
certain
type]
40
SHINICHI
MOCHIZUKI
—
cf.
the
discussion
of
[IUTchII],
Remark
3.6.3,
(i);
[AbsTopIII],
Remark
3.7.7,
(i).
In
this
context,
it
is
useful
to
recall
that
étale
fundamental
groups
—
i.e.,
Galois
groups
—
are
defined
as
certain
automorphism
groups
of
fields/rings;
in
particular,
the
definition
of
such
a
Galois
group
“as
a
certain
automorphism
group
of
some
ring
structure”
is
incompatible,
in
a
quite
essential
way,
with
the
log-wall.
In
a
similar
vein,
Kummer
theory,
which
depends
on
the
multiplicative
structure
of
the
ring
under
consideration,
is
also
incompatible,
in
a
quite
essential
way,
with
the
log-wall
[cf.
Proposition
1.2,
(iv)].
That
is
to
say,
in
the
context
of
the
log-link,
the
only
structure
of
interest
that
is
manifestly
compatible
with
the
log-
link
[cf.
Proposition
1.2,
(i),
(ii)]
is
the
associated
D-prime-strip
—
i.e.,
the
abstract
topological
groups
[isomorphic
to
“Π
v
”
—
cf.
the
notation
of
[IUTchI],
Definition
3.1,
(e),
(f)]
at
v
∈
V
non
and
abstract
Aut-holomorphic
spaces
[isomorphic
to
“U
v
”
—
cf.
the
notation
of
[IUTchII],
Proposition
4.3]
at
v
∈
V
arc
.
Indeed,
this
observation
is
precisely
the
starting
point
of
the
theory
of
[AbsTopIII]
[cf.
the
discussion
of
[AbsTopIII],
§I1,
§I4].
(ii)
Other
important
examples
of
structures
which
are
incompatible
with
the
log-wall
include
(a)
the
additive
structure
on
the
image
of
the
Kummer
map
[cf.
the
discussion
of
[AbsTopIII],
Remark
3.7.5];
(b)
in
the
“birational”
situation
—
i.e.,
where
one
replaces
“Π
v
”
by
the
abso-
of
the
function
field
of
the
affine
curve
that
gave
lute
Galois
group
Π
birat
v
rise
to
Π
v
—
the
datum
of
the
collection
of
closed
points
that
determines
the
affine
curve
[cf.
[AbsTopIII],
Remark
3.7.7,
(ii)].
Note,
for
instance
in
the
case
of
(b),
when,
say,
for
simplicity,
v
∈
V
good
V
non
,
that
one
may
think
of
the
additional
datum
under
consideration
as
consisting
of
Π
v
that
arises
from
the
scheme-theoretic
the
natural
outer
surjection
Π
birat
v
morphism
from
the
spectrum
of
the
function
field
to
the
given
affine
curve.
On
the
other
hand,
just
as
in
the
case
of
the
discussion
of
scheme-theoretic
basepoints
in
(i),
the
construction
of
such
an
object
Π
birat
Π
v
whose
structure
depends,
in
an
v
essential
way,
on
the
scheme
[i.e.,
ring!]
structures
involved
necessarily
fails
to
be
compatible
with
the
log-link
[cf.
the
discussion
of
[AbsTopIII],
Remark
3.7.7,
(ii)].
(iii)
One
way
to
understand
the
incompatibility
discussed
in
(ii),
(b),
is
as
,
Δ
v
for
the
respective
kernels
of
the
natural
surjections
follows.
Write
Δ
birat
v
birat
Π
v
G
v
,
Π
v
G
v
.
Then
if
one
forgets
about
the
scheme-theoretic
base-
points
discussed
in
(i),
G
v
,
Δ
birat
,
and
Δ
v
may
be
understood
on
both
sides
of
v
,
the
log-wall
as
“some
topological
group”,
and
each
of
the
topological
groups
Δ
birat
v
Δ
v
may
be
understood
on
both
sides
of
the
log-wall
as
being
equipped
with
“some
outer
G
v
-action”
—
cf.
the
two
diagonal
arrows
of
Fig.
1.2
below.
On
the
other
Δ
v
[cf.
the
dotted
line
in
hand,
the
datum
of
a
particular
outer
surjection
Δ
birat
v
Fig.
1.2]
relating
these
two
diagonal
arrows
—
which
depends,
in
an
essential
way,
on
the
scheme
[i.e.,
ring]
structures
involved!
—
necessarily
fails
to
be
compatible
with
the
log-link
[cf.
the
discussion
of
[AbsTopIII],
Remark
3.7.7,
(ii)].
This
issue
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
41
of
“triangular
compatibility
between
independent
indeterminacies”
is
formally
remi-
niscent
of
the
issue
of
compatibility
of
outer
homomorphisms
discussed
in
[IUTchI],
Remark
4.5.1,
(i)
[cf.
also
[IUTchII],
Remark
2.5.2,
(ii)].
indep.
bp.
indet.
nonarch.
local
abs.
Galois
group
G
v
indep.
bp.
indet.
birational
geom.
fund.
gp.
Δ
birat
v
?
...............
indep.
bp.
indet.
affine
geom.
fund.
gp.
Δ
v
Fig.
1.2:
Independent
basepoint
indeterminacies
obstruct
relationship
between
birational
and
affine
geometric
fundamental
groups
Remark
1.2.5.
The
discussion
in
Remark
1.2.4
of
the
incompatibility
of
the
log-
wall
with
various
structures
that
arise
from
ring/scheme-theory
is
closely
related
to
the
issue
of
avoiding
the
use
of
fixed
ring/scheme-theoretic
reference
mod-
els
in
mono-anabelian
construction
algorithms
[cf.
the
discussion
of
[IUTchI],
Remark
3.2.1,
(i);
[AbsTopIII],
§I4].
Put
another
way,
at
least
in
the
context
of
the
log-link
[i.e.,
situations
of
the
sort
considered
in
[AbsTopIII],
as
well
as
in
the
present
paper],
mono-anabelian
construction
algorithms
may
be
understood
as
algorithms
whose
dependence
on
data
arising
from
such
fixed
ring/scheme-
theoretic
reference
models
is
“invariant”,
or
“coric”,
with
respect
to
the
action
of
log
on
such
models.
A
substantial
portion
of
[AbsTopIII],
§3,
is
devoted
precisely
to
the
task
of
giving
a
precise
formulation
of
this
concept
of
“invariance”
by
means
of
such
notions
as
observables,
families
of
homotopies,
and
telecores.
For
instance,
one
approach
to
formulating
the
failure
of
the
ring
structure
of
a
fixed
reference
model
to
be
“coric”
with
respect
to
log
may
be
seen
in
[AbsTopIII],
Corollary
3.6,
(iv);
[AbsTopIII],
Corollary
3.7,
(iv).
Proposition
1.3.
(log-links
Between
Θ
±ell
NF-Hodge
Theaters)
Let
†
HT
Θ
±ell
NF
;
‡
HT
Θ
±ell
NF
42
SHINICHI
MOCHIZUKI
be
Θ
±ell
NF-Hodge
theaters
[relative
to
the
given
initial
Θ-data]
—
cf.
[IUTchI],
±ell
±ell
Definition
6.13,
(i).
Write
†
HT
D-Θ
NF
,
‡
HT
D-Θ
NF
for
the
associated
D-
Θ
±ell
NF-Hodge
theaters
—
cf.
[IUTchI],
Definition
6.13,
(ii).
Then:
(i)
(Construction
of
the
log-Link)
Fix
an
isomorphism
±ell
Ξ
:
†
HT
D-Θ
∼
‡
→
NF
HT
D-Θ
±ell
NF
of
D-Θ
±ell
NF-Hodge
theaters.
Let
†
F
be
one
of
the
F-prime-strips
that
appear
±ell
in
the
Θ-
and
Θ
±
-bridges
that
constitute
†
HT
Θ
NF
—
i.e.,
either
one
of
the
F-prime-strips
†
F
>
,
†
F
or
one
of
the
constituent
F-prime-strips
of
the
capsules
†
†
F
J
,
F
T
[cf.
[IUTchI],
Definition
5.5,
(ii);
[IUTchI],
Definition
6.11,
(i)].
Write
‡
F
for
±ell
the
corresponding
F-prime-strip
of
‡
HT
Θ
NF
.
Then
the
poly-isomorphism
deter-
mined
by
Ξ
between
the
D-prime-strips
associated
to
†
F
,
‡
F
uniquely
determines
∼
a
poly-isomorphism
log(
†
F
)
→
‡
F
[cf.
Definition
1.1,
(iii);
[IUTchI],
Corol-
log
lary
5.3,
(ii)],
hence
a
log-link
†
F
denote
by
†
HT
Θ
±ell
−→
NF
‡
F
[cf.
Definition
1.1,
(iii)].
We
shall
‡
log
−→
±ell
HT
Θ
±ell
NF
±ell
and
refer
to
as
a
log-link
from
†
HT
Θ
NF
to
‡
HT
Θ
NF
the
collection
of
data
log
consisting
of
Ξ,
together
with
the
collection
of
log-links
†
F
−→
‡
F
,
as
“”
ranges
over
all
possibilities
for
the
F-prime-strips
in
question.
When
Ξ
is
replaced
±ell
±ell
∼
by
a
poly-isomorphism
†
HT
D-Θ
NF
→
‡
HT
D-Θ
NF
,
we
shall
also
refer
to
the
resulting
collection
of
log-links
[i.e.,
corresponding
to
each
constituent
isomorphism
±ell
±ell
of
the
poly-isomorphism
Ξ]
as
a
log-link
from
†
HT
Θ
NF
to
‡
HT
Θ
NF
.
When
Ξ
is
the
full
poly-isomorphism,
we
shall
refer
to
the
resulting
log-link
as
the
full
log-link.
±ell
log
(ii)
(Coricity)
Any
log-link
†
HT
Θ
NF
−→
‡
HT
Θ
be
thought
of
as
“lying
over”]
a
[poly-]isomorphism
†
±ell
HT
D-Θ
NF
∼
→
‡
±ell
HT
D-Θ
±ell
NF
induces
[and
may
NF
of
D-Θ
±ell
NF-Hodge
theaters
[and
indeed
coincides
with
the
log-link
constructed
in
(i)
from
this
[poly-]isomorphism
of
D-Θ
±ell
NF-Hodge
theaters].
(iii)
(Further
Properties
of
the
log-Link)
In
the
notation
of
(i),
any
log-
±ell
log
±ell
link
†
HT
Θ
NF
−→
‡
HT
Θ
NF
satisfies,
for
each
F-prime-strip
†
F
,
properties
corresponding
to
the
properties
of
Proposition
1.2,
(ii),
(iii),
(iv),
(v),
(vi),
(vii),
(viii),
(ix),
i.e.,
concerning
simultaneous
compatibility
with
ring
structures
and
log-volumes,
Kummer
theory,
and
log-shells.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
43
±ell
(iv)
(Frobenius-picture)
Let
{
n
HT
Θ
NF
}
n∈Z
be
a
collection
of
distinct
Θ
±ell
NF-Hodge
theaters
[relative
to
the
given
initial
Θ-data]
indexed
by
the
in-
±ell
tegers.
Write
{
n
HT
D-Θ
NF
}
n∈Z
for
the
associated
D-Θ
±ell
NF-Hodge
theaters.
Then
the
full
log-links
n
HT
Θ
to
an
infinite
chain
log
.
.
.
−→
(n−1)
±ell
HT
Θ
NF
±ell
log
−→
NF
n
log
−→
HT
Θ
(n+1)
±ell
NF
HT
Θ
log
−→
±ell
NF
(n+1)
,
for
n
∈
Z,
give
rise
±ell
HT
Θ
NF
log
−→
.
.
.
of
log-linked
Θ
±ell
NF-Hodge
theaters
which
induces
a
chain
of
full
poly-isomor-
phisms
±ell
±ell
∼
∼
∼
.
.
.
→
n
HT
D-Θ
NF
→
(n+1)
HT
D-Θ
NF
→
.
.
.
on
the
associated
D-Θ
±ell
NF-Hodge
theaters.
These
chains
may
be
represented
symbolically
as
an
oriented
graph
Γ
[cf.
[AbsTopIII],
§0]
...
→
...
•
→
•
↓
→
•
→
...
...
◦
log
—
i.e.,
where
the
horizontal
arrows
correspond
to
the
“
−→
’s”;
the
“•’s”
corre-
±ell
±ell
spond
to
the
“
n
HT
Θ
NF
”;
the
“◦”
corresponds
to
the
“
n
HT
D-Θ
NF
”,
identified
up
to
isomorphism;
the
vertical/diagonal
arrows
correspond
to
the
Kummer
iso-
morphisms
implicit
in
the
statement
of
(iii).
This
oriented
graph
Γ
admits
a
natural
action
by
Z
[cf.
[AbsTopIII],
Corollary
5.5,
(v)]
—
i.e.,
a
translation
symmetry
—
that
fixes
the
“core”
◦,
but
it
does
not
admit
arbitrary
per-
mutation
symmetries.
For
instance,
Γ
does
not
admit
an
automorphism
that
switches
two
adjacent
vertices,
but
leaves
the
remaining
vertices
fixed.
Proof.
The
various
assertions
of
Proposition
1.3
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
1.3.1.
Note
that
in
Proposition
1.3,
(i),
it
was
necessary
to
carry
out
the
given
construction
of
the
log-link
first
for
a
single
Ξ
[i.e.,
as
opposed
to
a
poly-isomorphism
Ξ],
in
order
to
maintain
compatibility
with
the
crucial
“±-
synchronization”
[cf.
[IUTchI],
Remark
6.12.4,
(iii);
[IUTchII],
Remark
4.5.3,
(iii)]
inherent
in
the
structure
of
a
Θ
±ell
-Hodge
theater.
Remark
1.3.2.
In
the
construction
of
Proposition
1.3,
(i),
the
constituent
F-
prime-strips
†
F
t
,
for
t
∈
T
,
of
the
capsule
†
F
T
are
considered
without
regard
to
the
F
±
l
-symmetries
discussed
in
[IUTchII],
Corollary
4.6,
(iii).
On
the
other
hand,
one
verifies
immediately
that
the
log-links
associated,
in
the
construction
of
Proposi-
tion
1.3,
(i),
to
these
F-prime-strips
†
F
t
,
for
t
∈
T
—
i.e.,
more
precisely,
associated
to
the
labeled
collections
of
monoids
Ψ
cns
(
†
F
)
t
of
[IUTchII],
Corollary
4.6,
(iii)
—
are
in
fact
compatible
with
the
F
±
l
-symmetrizing
isomorphisms
discussed
in
[IUTchII],
Corollary
4.6,
(iii),
hence
also
with
the
conjugate
synchronization
determined
by
these
F
±
l
-symmetrizing
isomorphisms
—
cf.
the
discussion
of
Step
44
SHINICHI
MOCHIZUKI
(vi)
of
the
proof
of
Corollary
3.12
of
§3
below.
We
leave
the
routine
details
to
the
reader.
Remark
1.3.3.
(i)
In
the
context
of
Proposition
1.3
[cf.
also
the
discussion
of
Remarks
1.2.4,
1.3.1,
1.3.2],
it
is
of
interest
to
observe
that
the
relationship
between
the
various
Frobenioid-theoretic
[i.e.,
Frobenius-like!]
portions
of
the
Θ
±ell
NF-Hodge
the-
aters
in
the
domain
and
codomain
of
the
log-link
of
Proposition
1.3,
(i),
does
not
include
any
data
—
i.e.,
of
the
sort
discussed
in
Remark
1.2.4,
(ii),
(a),
(b);
Remark
1.2.4,
(iii)
—
that
is
incompatible,
relative
to
the
relevant
Kummer
isomorphisms,
with
the
coricity
property
for
étale-
like
structures
given
in
Proposition
1.3,
(ii).
Indeed,
this
observation
may
be
understood
as
a
consequence
of
the
fact
[cf.
Re-
marks
1.3.1,
1.3.2;
[IUTchI],
Corollary
5.3,
(i),
(ii),
(iv);
[IUTchI],
Corollary
5.6,
(i),
(ii),
(iii)]
that
these
Frobenioid-theoretic
portions
of
the
Θ
±ell
NF-Hodge
the-
aters
under
consideration
are
completely
[i.e.,
fully
faithfully!]
controlled
[cf.
the
discussion
of
(ii)
below
for
more
details],
via
functorial
algorithms,
by
the
cor-
responding
étale-like
structures,
i.e.,
structures
that
appear
in
the
associated
D-
Θ
±ell
NF-Hodge
theaters,
which
satisfy
the
crucial
coricity
property
of
Proposition
1.3,
(ii).
(ii)
In
the
context
of
(i),
it
is
of
interest
to
recall
that
the
global
portion
of
the
underlying
Θ
ell
-bridges
is
defined
[cf.
[IUTchI],
Definition
6.11,
(ii)]
in
such
a
way
that
it
does
not
contain
any
global
Frobenioid-theoretic
data!
In
particular,
the
issue
discussed
in
(i)
concerns
only
the
Frobenioid-theoretic
portions
of
the
following:
(a)
the
various
F-prime-strips
that
appear;
(b)
the
underlying
Θ-Hodge
theaters
of
the
Θ
±ell
NF-Hodge
theaters
under
consideration;
(c)
the
global
portion
of
the
underlying
NF-bridges
of
the
Θ
±ell
NF-Hodge
theaters
under
consideration.
Here,
the
Frobenioid-theoretic
data
of
(c)
gives
rise
to
independent
[i.e.,
for
cor-
responding
portions
of
the
Θ
±ell
NF-Hodge
theaters
in
the
domain
and
codomain
of
the
log-link]
basepoints
with
respect
to
the
F
l
-symmetry
[cf.
[IUTchI],
Corol-
lary
5.6,
(iii);
[IUTchI],
Remark
6.12.6,
(iii);
[IUTchII],
Remark
4.7.6].
On
the
other
hand,
the
independent
basepoints
that
arise
from
the
Frobenioid-theoretic
data
of
(b),
as
well
as
of
the
portion
of
(a)
that
lies
in
the
underlying
ΘNF-Hodge
theater,
do
not
cause
any
problems
[i.e.,
from
the
point
of
view
of
the
sort
of
in-
compatibility
discussed
in
(i)]
since
this
data
is
only
subject
to
relationships
defined
by
means
of
full
poly-isomorphisms
[cf.
[IUTchI],
Examples
4.3,
4.4].
That
is
to
say,
the
F-prime-strips
that
lie
in
the
underlying
Θ
±ell
-Hodge
theater
constitute
the
most
delicate
[i.e.,
relative
to
the
issue
of
independent
basepoints!]
portion
of
the
Frobenioid-theoretic
data
of
a
Θ
±ell
NF-Hodge
theater.
This
delicacy
revolves
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
45
around
the
global
synchronization
of
±-indeterminacies
in
the
underlying
Θ
±ell
-
Hodge
theater
[cf.
[IUTchI],
Remark
6.12.4,
(iii);
[IUTchII],
Remark
4.5.3,
(iii)].
On
the
other
hand,
this
delicacy
does
not
in
fact
cause
any
problems
[i.e.,
from
the
point
of
view
of
the
sort
of
incompatibility
discussed
in
(i)]
since
[cf.
[IUTchI],
Remark
6.12.4,
(iii);
[IUTchII],
Remark
4.5.3,
(iii)]
the
synchronizations
of
±-
indeterminacies
in
the
underlying
Θ
±ell
-Hodge
theater
are
defined
[not
by
means
of
scheme-theoretic
relationships,
but
rather]
by
applying
the
intrinsic
structure
of
the
underlying
D-Θ
±ell
-Hodge
theater,
which
satisfies
the
crucial
coricity
property
of
Proposition
1.3,
(ii)
[cf.
the
discussion
of
(i);
Remarks
1.3.1,
1.3.2].
The
diagrams
discussed
in
the
following
Definition
1.4
will
play
a
central
role
in
the
theory
of
the
present
series
of
papers.
Definition
1.4.
We
maintain
the
notation
of
Proposition
1.3
[cf.
also
[IUTchII],
±ell
Corollary
4.10,
(iii)].
Let
{
n,m
HT
Θ
NF
}
n,m∈Z
be
a
collection
of
distinct
Θ
±ell
NF-
Hodge
theaters
[relative
to
the
given
initial
Θ-data]
indexed
by
pairs
of
integers.
Then
we
shall
refer
to
either
of
the
diagrams
..
..
.
.
⏐
log
⏐
log
⏐
⏐
...
...
...
...
Θ
×μ
n,m+1
Θ
×μ
n,m
−→
−→
Θ
×μ
gau
−→
Θ
×μ
gau
−→
NF
±ell
HT
Θ
NF
⏐
log
⏐
n,m+1
n,m
±ell
HT
Θ
⏐
log
⏐
Θ
×μ
n+1,m+1
Θ
×μ
n+1,m
−→
−→
HT
Θ
⏐
log
⏐
±ell
HT
Θ
⏐
log
⏐
..
.
..
.
..
.
⏐
log
⏐
..
.
⏐
log
⏐
±ell
HT
Θ
⏐
log
⏐
NF
Θ
±ell
NF
HT
⏐
log
⏐
..
.
Θ
×μ
gau
−→
Θ
×μ
gau
−→
n+1,m+1
n+1,m
±ell
HT
Θ
⏐
log
⏐
NF
NF
±ell
NF
Θ
±ell
NF
HT
⏐
log
⏐
Θ
×μ
−→
Θ
×μ
−→
Θ
×μ
gau
−→
Θ
×μ
gau
−→
...
...
...
...
..
.
—
where
the
vertical
arrows
are
the
full
log-links,
and
the
horizontal
arrows
are
the
Θ
×μ
-
and
Θ
×μ
gau
-links
of
[IUTchII],
Corollary
4.10,
(iii)
—
as
the
log-theta-lattice.
We
shall
refer
to
the
log-theta-lattice
that
involves
the
Θ
×μ
-
(respectively,
Θ
×μ
gau
-)
46
SHINICHI
MOCHIZUKI
links
as
non-Gaussian
(respectively,
Gaussian).
Thus,
either
of
these
diagrams
may
be
represented
symbolically
by
an
oriented
graph
..
.
⏐
⏐
..
.
⏐
⏐
...
−→
•
⏐
⏐
−→
•
⏐
⏐
−→
.
.
.
...
−→
•
⏐
⏐
−→
•
⏐
⏐
−→
.
.
.
..
.
..
.
±ell
—
where
the
“•’s”
correspond
to
the
“
n,m
HT
Θ
NF
”.
Remark
1.4.1.
(i)
One
fundamental
property
of
the
log-theta-lattices
discussed
in
Definition
1.4
is
the
following:
the
various
squares
that
appear
in
each
of
the
log-theta-lattices
discussed
in
Definition
1.4
are
far
from
being
[1-]commutative!
Indeed,
whereas
the
vertical
arrows
in
each
log-theta-lattice
are
constructed
by
applying
the
various
logarithms
at
v
∈
V
—
i.e.,
which
are
defined
by
means
of
power
series
that
depend,
in
an
essential
way,
on
the
local
ring
structures
at
v
∈
V
—
the
horizontal
arrows
in
each
log-theta-lattice
[i.e.,
the
Θ
×μ
-,
Θ
×μ
gau
-links]
are
incompatible
with
these
local
ring
structures
at
v
∈
V
in
an
essential
way
[cf.
[IUTchII],
Remark
1.11.2,
(i),
(ii)].
(ii)
Whereas
the
horizontal
arrows
in
each
log-theta-lattice
[i.e.,
the
Θ
×μ
-,
×μ
’s”
at
[for
Θ
×μ
gau
-links]
allow
one,
roughly
speaking,
to
identify
the
respective
“O
non
simplicity]
v
∈
V
on
either
side
of
the
horizontal
arrow
[cf.
[IUTchII],
Corollary
4.10,
(iv)],
in
order
to
avail
oneself
of
the
theory
of
log-shells
—
which
will
play
an
essential
role
in
the
multiradial
representation
of
the
Gaussian
monoids
to
be
developed
in
§3
below
—
it
is
necessary
for
the
“•”
[i.e.,
Θ
±ell
NF-Hodge
theater]
in
which
one
operates
to
appear
as
the
codomain
of
a
log-link,
i.e.,
of
a
vertical
arrow
of
the
log-theta-lattice
[cf.
the
discussion
of
[AbsTopIII],
Remark
5.10.2,
(iii)].
That
is
to
say,
from
the
point
of
view
of
the
goal
of
constructing
the
multiradial
representation
of
the
Gaussian
monoids
that
is
to
be
developed
in
§3
below,
each
execution
of
a
horizontal
arrow
of
the
log-theta-lattice
necessarily
obligates
a
subsequent
execution
of
a
vertical
arrow
of
the
log-theta-lattice.
On
the
other
hand,
in
light
of
the
noncommutativity
observed
in
(i),
this
“in-
tertwining”
of
the
horizontal
and
vertical
arrows
of
the
log-theta-lattice
means
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
47
that
the
desired
multiradiality
—
i.e.,
simultaneous
compatibility
with
the
arithmetic
holomorphic
structures
on
both
sides
of
a
horizontal
arrow
of
the
log-
theta-lattice
—
can
only
be
realized
[cf.
the
discussion
of
Remark
1.2.2,
(iii)]
if
one
works
with
objects
that
are
invariant
with
respect
to
the
vertical
arrows
[i.e.,
with
respect
to
the
action
of
Z
discussed
in
Proposition
1.3,
(iv)],
that
is
to
say,
with
“vertical
cores”,
of
the
log-theta-lattice.
(iii)
From
the
point
of
view
of
the
analogy
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory
[cf.
[AbsTopIII],
§I5],
the
vertical
arrows
of
the
log-theta-lattice
correspond
to
the
Frobenius
morphism
in
positive
characteristic,
whereas
the
horizontal
arrows
of
the
log-theta-lattice
correspond
to
the
“transition
from
p
n
Z/p
n+1
Z
to
p
n−1
Z/p
n
Z”,
i.e.,
the
mixed
characteristic
extension
structure
of
a
ring
of
Witt
vectors
[cf.
[IUTchI],
Remark
3.9.3,
(i)].
These
correspondences
are
summarized
in
Fig.
1.3
below.
In
particular,
the
“intertwining
of
horizontal
and
vertical
arrows
of
the
log-theta-lattice”
discussed
in
(ii)
above
may
be
thought
of
as
the
analogue,
in
the
context
of
the
theory
of
the
present
series
of
papers,
of
the
well-known
“intertwining
between
the
mixed
characteristic
extension
structure
of
a
ring
of
Witt
vectors
and
the
Frobenius
morphism
in
positive
characteristic”
that
appears
in
the
classical
p-adic
theory.
horizontal
arrows
of
the
log-theta-lattice
mixed
characteristic
extension
structure
of
a
ring
of
Witt
vectors
vertical
arrows
of
the
log-theta-lattice
the
Frobenius
morphism
in
positive
characteristic
Fig.
1.3:
Analogy
between
the
log-theta-lattice
and
p-adic
Teichmüller
theory
Remark
1.4.2.
(i)
The
horizontal
and
vertical
arrows
of
the
log-theta-lattices
discussed
in
Definition
1.4
share
the
common
property
of
being
incompatible
with
the
local
ring
structures,
hence,
in
particular,
with
the
conventional
scheme-theoretic
basepoints
on
either
side
of
the
arrow
in
question
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.3,
(i)].
On
the
other
hand,
whereas
the
linking
data
of
the
vertical
arrows
[i.e.,
the
log-
link]
is
rigid
and
corresponds
to
a
single
fixed,
rigid
arithmetic
holomorphic
structure
in
which
addition
and
multiplication
are
subject
to
“rotations”
[cf.
the
discussion
of
[AbsTopIII],
§I3],
the
linking
data
of
the
horizontal
arrows
[i.e.,
the
×μ
’s”
at
[for
simplicity]
v
∈
V
non
Θ
×μ
-,
Θ
×μ
gau
-links]
—
i.e.,
more
concretely,
the
“O
×
-indeterminacy,
which
has
the
effect
of
obliterating
the
—
is
subject
to
a
Z
arithmetic
holomorphic
structure
associated
to
a
vertical
line
of
the
log-theta-lattice
[cf.
the
discussion
of
[IUTchII],
Remark
1.11.2,
(i),
(ii)].
(ii)
If,
in
the
spirit
of
the
discussion
of
[IUTchII],
Remark
1.11.2,
(ii),
one
attempts
to
“force”
the
horizontal
arrows
of
the
log-theta-lattice
to
be
compat-
ible
with
the
arithmetic
holomorphic
structures
on
either
side
of
the
arrow
by
48
SHINICHI
MOCHIZUKI
declaring
—
in
the
style
of
the
log-link!
—
that
these
horizontal
arrows
induce
an
isomorphism
of
the
respective
“Π
v
’s”
at
[for
simplicity]
v
∈
V
non
,
then
one
must
contend
with
a
situation
in
which
the
“common
arithmetic
holomorphic
structure
rigidified
by
the
isomorphic
copies
of
Π
v
”
is
obliterated
each
time
one
takes
into
×
-
×
[i.e.,
that
arises
from
the
Z
account
the
action
of
a
nontrivial
element
of
Z
indeterminacy
involved]
on
the
corresponding
“O
×μ
”.
In
particular,
in
order
to
keep
track
of
the
arithmetic
holomorphic
structure
currently
under
consideration,
one
must,
in
effect,
consider
paths
that
record
the
sequence
of
“Π
v
-rigidifying”
×
-indeterminacy”
operations
that
one
invokes.
On
the
other
hand,
the
hor-
and
“
Z
izontal
lines
of
the
log-theta-lattices
given
in
Definition
1.4
amount,
in
effect,
to
universal
covering
spaces
of
the
loops
—
i.e.,
“unraveling
paths
of
the
loops”
[cf.
the
discussion
of
Remark
1.2.2,
(vi)]
—
that
occur
as
one
invokes
various
series
×
-indeterminacy”
operations.
Thus,
in
summary,
any
of
“Π
v
-rigidifying”
and
“
Z
attempt
as
described
above
to
“force”
the
horizontal
arrows
of
the
log-theta-lattice
to
be
compatible
with
the
arithmetic
holomorphic
structures
on
either
side
of
the
arrow
does
not
result
in
any
substantive
simplification
of
the
theory
of
the
present
series
of
papers.
We
refer
the
reader
to
[IUTchIV],
Remark
3.6.3,
for
a
discussion
of
a
related
topic.
We
are
now
ready
to
state
the
main
result
of
the
present
§1.
(Bi-cores
of
the
Log-theta-lattice)
Fix
a
collection
of
initial
Theorem
1.5.
Θ-data
(F
/F,
X
F
,
l,
C
K
,
V,
V
bad
mod
,
)
as
in
[IUTchI],
Definition
3.1.
Then
any
Gaussian
log-theta-lattice
correspond-
ing
to
this
collection
of
initial
Θ-data
[cf.
Definition
1.4]
satisfies
the
following
properties:
(i)
(Vertical
Coricity)
The
vertical
arrows
of
the
Gaussian
log-theta-lattice
induce
full
poly-isomorphisms
between
the
respective
associated
D-Θ
±ell
NF-Hodge
theaters
∼
...
→
n,m
±ell
HT
D-Θ
NF
∼
→
n,m+1
±ell
HT
D-Θ
NF
∼
→
...
[cf.
Proposition
1.3,
(ii)].
Here,
n
∈
Z
is
held
fixed,
while
m
∈
Z
is
allowed
to
vary.
(ii)
(Horizontal
Coricity)
The
horizontal
arrows
of
the
Gaussian
log-theta-
lattice
induce
full
poly-isomorphisms
between
the
respective
associated
F
×μ
-
prime-strips
...
∼
→
n,m
×μ
F
∼
→
n+1,m
×μ
F
∼
→
...
[cf.
[IUTchII],
Corollary
4.10,
(iv)].
Here,
m
∈
Z
is
held
fixed,
while
n
∈
Z
is
allowed
to
vary.
(iii)
(Bi-coric
F
×μ
-Prime-Strips)
For
n,
m
∈
Z,
write
n,m
D
for
the
D
-
prime-strip
associated
to
the
F
-prime-strip
n,m
F
labeled
“”
of
the
Θ
±ell
NF-
±ell
Hodge
theater
n,m
HT
Θ
NF
[cf.
[IUTchII],
Corollary
4.10,
(i)];
n,m
D
for
the
D-
±ell
prime-strip
labeled
“”
of
the
Θ
±ell
NF-Hodge
theater
n,m
HT
Θ
NF
[cf.
[IUTchI],
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
49
Definitions
6.11,
(i),
(iii);
6.13,
(i)].
Let
us
identify
[cf.
[IUTchII],
Corollary
4.10,
(i)]
the
collections
of
data
Ψ
cns
(
n,m
D
)
0
and
Ψ
cns
(
n,m
D
)
F
l
via
the
isomorphism
of
the
final
display
of
[IUTchII],
Corollary
4.5,
(iii),
and
denote
by
F
(
n,m
D
)
the
resulting
F
-prime-strip.
[Thus,
it
follows
immediately
from
the
constructions
involved
—
cf.
the
discussion
of
[IUTchII],
Corollary
4.10,
(i)
—
that
there
is
a
∼
natural
identification
isomorphism
F
(
n,m
D
)
→
F
>
(
n,m
D
>
),
where
we
write
F
>
(
n,m
D
>
)
for
the
F
-prime-strip
determined
by
Ψ
cns
(
n,m
D
>
).]
Write
F
×μ
(
n,m
D
)
F
×
(
n,m
D
),
for
the
F
×
-,
F
×μ
-prime-strips
determined
by
F
(
n,m
D
)
[cf.
[IUTchII],
Def-
∼
→
inition
4.9,
(vi),
(vii)].
Thus,
by
applying
the
isomorphisms
“Ψ
cns
(
‡
D)
×
v
‡
×
Ψ
ss
cns
(
D
)
v
”,
for
v
∈
V,
of
[IUTchII],
Corollary
4.5,
(ii),
[it
follows
immediately
from
the
definitions
that]
there
exists
a
functorial
algorithm
in
the
D
-prime-
strip
n,m
D
for
constructing
an
F
×
-prime-strip
F
×
(
n,m
D
),
together
with
a
functorial
algorithm
in
the
D-prime-strip
n,m
D
for
constructing
a
natural
isomorphism
∼
F
×
(
n,m
D
)
→
F
×
(
n,m
D
)
—
i.e.,
in
more
intuitive
terms,
“F
×
(
n,m
D
)”,
hence
also
the
associated
F
×μ
-
prime-strip
“F
×μ
(
n,m
D
)”,
may
be
naturally
regarded,
up
to
isomorphism,
as
objects
constructed
from
n,m
D
.
Then
the
poly-isomorphisms
of
(i)
[cf.
Remark
1.3.2],
(ii)
induce,
respectively,
poly-isomorphisms
of
F
×μ
-prime-strips
∼
∼
∼
∼
∼
∼
.
.
.
→
F
×μ
(
n,m
D
)
→
F
×μ
(
n,m+1
D
)
→
.
.
.
.
.
.
→
F
×μ
(
n,m
D
)
→
F
×μ
(
n+1,m
D
)
→
.
.
.
—
where
we
note
that,
relative
to
the
natural
isomorphisms
of
F
×μ
-prime-strips
∼
F
×
(
n,m
D
)
→
F
×
(
n,m
D
)
discussed
above,
the
collection
of
isomorphisms
that
constitute
the
poly-isomorphisms
of
F
×μ
-prime-strips
of
the
first
line
of
the
display
is,
in
general,
strictly
smaller
than
the
collection
of
isomorphisms
that
constitute
the
poly-isomorphisms
of
F
×μ
-prime-strips
of
the
second
line
of
the
display
[cf.
the
existence
of
non-scheme-theoretic
automorphisms
of
absolute
Galois
groups
of
MLF’s,
as
discussed
in
[AbsTopIII],
§I3];
the
poly-isomorphisms
of
F
×μ
-prime-
strips
of
the
second
line
of
the
display
are
not
full
[cf.
[IUTchII],
Remark
1.8.1].
In
particular,
by
composing
these
isomorphisms,
one
obtains
poly-isomorphisms
of
F
×μ
-prime-strips
∼
F
×μ
(
n,m
D
)
→
F
×μ
(
n
,m
D
)
for
arbitrary
n
,
m
∈
Z.
That
is
to
say,
in
more
intuitive
terms,
the
F
×μ
-prime-
strip
“
n,m
F
×μ
(
n,m
D
)”,
regarded
up
to
a
certain
class
of
isomorphisms,
is
an
50
SHINICHI
MOCHIZUKI
invariant
—
which
we
shall
refer
to
as
“bi-coric”
—
of
both
the
horizontal
and
the
vertical
arrows
of
the
Gaussian
log-theta-lattice.
Finally,
the
Kummer
iso-
∼
morphisms
“Ψ
cns
(
‡
F)
→
Ψ
cns
(
‡
D)”
of
[IUTchII],
Corollary
4.6,
(i),
determine
Kummer
isomorphisms
n,m
×μ
F
∼
→
F
×μ
(
n,m
D
)
which
are
compatible
with
the
poly-isomorphisms
of
(ii),
as
well
as
with
the
×μ-
Kummer
structures
at
the
v
∈
V
non
of
the
various
F
×μ
-prime-strips
involved
[cf.
[IUTchII],
Definition
4.9,
(vi),
(vii)];
a
similar
compatibility
holds
for
v
∈
V
arc
[cf.
the
discussion
of
the
final
portion
of
[IUTchII],
Definition
4.9,
(v)].
(iv)
(Bi-coric
Mono-analytic
Log-shells)
The
poly-isomorphisms
that
con-
stitute
the
bi-coricity
property
discussed
in
(iii)
induce
poly-isomorphisms
∼
I
n,m
D
⊆
log(
n,m
D
)
→
I
n
,m
D
⊆
log(
n
,m
D
)
I
F
×μ
(
n,m
D
)
⊆
log(F
×μ
(
n,m
D
))
∼
→
I
F
×μ
(
n
,m
D
)
⊆
log(F
×μ
(
n
,m
D
))
for
arbitrary
n,
m,
n
,
m
∈
Z
that
are
compatible
with
the
natural
poly-isomor-
phisms
∼
I
n,m
D
⊆
log(
n,m
D
)
→
I
F
×μ
(
n,m
D
)
⊆
log(F
×μ
(
n,m
D
))
of
Proposition
1.2,
(viii).
On
the
other
hand,
by
applying
the
constructions
of
Definition
1.1,
(i),
(ii),
to
the
collections
of
data
“Ψ
cns
(
†
F
)
0
”
and
“Ψ
cns
(
†
F
)
F
”
l
used
in
[IUTchII],
Corollary
4.10,
(i),
to
construct
n,m
F
[cf.
Remark
1.3.2],
one
obtains
a
[“holomorphic”]
log-shell,
together
with
an
enveloping
“log(−)”
[cf.
the
pair
“I
†
F
⊆
log(
†
F)”
of
Definition
1.1,
(iii)],
which
we
denote
by
I
n,m
F
⊆
log(
n,m
F
)
[by
means
of
a
slight
abuse
of
notation,
since
no
F-prime-strip
“
n,m
F
”
has
been
defined!].
Then
one
has
natural
poly-isomorphisms
∼
I
n,m
D
⊆
log(
n,m
D
)
→
I
n,m
F
×μ
⊆
log(
n,m
F
×μ
)
∼
→
I
n,m
F
⊆
log(
n,m
F
)
∼
[cf.
the
poly-isomorphisms
obtained
in
Proposition
1.2,
(viii)];
here,
the
first
“
→
”
may
be
regarded
as
being
induced
by
the
Kummer
isomorphisms
of
(iii)
and
is
compatible
with
the
poly-isomorphisms
induced
by
the
poly-isomorphisms
of
(ii).
(v)
(Bi-coric
Mono-analytic
Global
Realified
Frobenioids)
Let
n,
m,
∼
n
,
m
∈
Z.
Then
the
poly-isomorphisms
of
D
-prime-strips
n,m
D
→
n
,m
D
induced
by
the
full
poly-isomorphisms
of
(i),
(ii)
induce
[cf.
[IUTchII],
Corollaries
4.5,
(ii);
4.10,
(v)]
an
isomorphism
of
collections
of
data
∼
(D
(
n,m
D
),
Prime(D
(
n,m
D
))
→
V,
{
n,m
ρ
D
,v
}
v∈V
)
∼
→
∼
(D
(
n
,m
D
),
Prime(D
(
n
,m
D
))
→
V,
{
n
,m
ρ
D
,v
}
v∈V
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
51
—
i.e.,
consisting
of
a
Frobenioid,
a
bijection,
and
a
collection
of
isomorphisms
of
topological
monoids
indexed
by
V.
Moreover,
this
isomorphism
of
collections
of
data
is
compatible,
relative
to
the
horizontal
arrows
of
the
Gaussian
log-theta-
lattice
[cf.,
e.g.,
the
full
poly-isomorphisms
of
(ii)],
with
the
R
>0
-orbits
of
the
isomorphisms
of
collections
of
data
∼
(
n,m
C
,
Prime(
n,m
C
)
→
V,
{
n,m
ρ
∼
→
,v
}
v∈V
)
∼
(D
(
n,m
D
),
Prime(D
(
n,m
D
))
→
V,
{
n,m
ρ
D
,v
}
v∈V
)
obtained
by
applying
the
functorial
algorithm
discussed
in
the
final
portion
of
[IUTchII],
Corollary
4.6,
(ii)
[cf.
also
the
latter
portions
of
[IUTchII],
Corollary
4.10,
(i),
(v)].
Proof.
The
various
assertions
of
Theorem
1.5
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
1.5.1.
(i)
Note
that
the
theory
of
conjugate
synchronization
developed
in
[IUTchII]
[cf.,
especially,
[IUTchII],
Corollaries
4.5,
(iii);
4.6,
(iii)]
plays
an
essential
role
in
establishing
the
bi-coricity
properties
discussed
in
Theorem
1.5,
(iii),
(iv),
(v)
—
i.e.,
at
a
more
technical
level,
in
constructing
the
objects
equipped
with
a
sub-
script
“”
that
appear
in
Theorem
1.5,
(iii);
[IUTchII],
Corollary
4.10,
(i).
That
is
to
say,
the
conjugate
synchronization
determined
by
the
various
symmetrizing
isomorphisms
of
[IUTchII],
Corollaries
4.5,
(iii);
4.6,
(iii),
may
be
thought
of
as
a
sort
of
descent
mechanism
that
allows
one
to
descend
data
that,
a
priori,
is
label-dependent
[i.e.,
depends
on
the
labels
“t
∈
LabCusp
±
(−)”]
to
data
that
is
label-independent.
Here,
it
is
important
to
recall
that
these
labels
depend,
in
an
essential
way,
on
the
“arithmetic
holomorphic
structures”
involved
—
i.e.,
at
a
more
technical
level,
on
the
geometric
fundamental
groups
involved
—
hence
only
make
sense
within
a
vertical
line
of
the
log-theta-lattice.
That
is
to
say,
the
significance
of
this
transition
from
label-dependence
to
label-independence
lies
in
the
fact
that
this
transition
is
precisely
what
allows
one
to
construct
objects
that
make
sense
in
horizontally
adjacent
“•’s”
of
the
log-theta-lattice,
i.e.,
to
construct
horizontally
coric
objects
[cf.
Theorem
1.5,
(ii);
the
second
line
of
the
fifth
display
of
Theorem
1.5,
(iii)].
On
the
other
hand,
in
order
to
construct
the
horizontal
arrows
of
the
log-theta-lattice,
it
is
necessary
to
work
with
Frobenius-like
struc-
tures
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.2,
(ii)].
In
particular,
in
order
to
construct
vertically
coric
objects
[cf.
the
first
line
of
the
fifth
display
of
Theo-
rem
1.5,
(iii)],
it
is
necessary
to
pass
to
étale-like
structures
[cf.
the
discussion
of
Remark
1.2.4,
(i)]
by
means
of
Kummer
isomorphisms
[cf.
the
final
display
of
Theorem
1.5,
(iii)].
Thus,
in
summary,
the
bi-coricity
properties
discussed
in
Theorem
1.5,
(iii),
(iv),
(v)
—
i.e.,
roughly
speaking,
the
bi-coricity
of
the
various
“O
×μ
”
at
v
∈
V
non
—
may
be
thought
of
as
a
consequence
of
the
intricate
interplay
of
various
aspects
of
the
theory
of
Kummer-compatible
conjugate
synchronization
es-
tablished
in
[IUTchII],
Corollaries
4.5,
(iii);
4.6,
(iii).
52
SHINICHI
MOCHIZUKI
(ii)
In
light
of
the
central
role
played
by
the
theory
of
conjugate
synchronization
in
the
constructions
that
underlie
Theorem
1.5
[cf.
the
discussion
of
(i)],
it
is
of
interest
to
examine
in
more
detail
to
what
extent
the
highly
technically
nontrivial
theory
of
conjugate
synchronization
may
be
replaced
by
a
simpler
apparatus.
One
naive
approach
to
this
problem
is
the
following.
Let
G
be
a
topological
group
[such
as
one
of
the
absolute
Galois
groups
G
v
associated
to
v
∈
V
non
].
Then
one
way
to
attempt
to
avoid
the
application
of
the
theory
of
conjugate
synchronization
—
which
amounts,
in
essence,
to
the
construction
of
a
diagonal
embedding
G
→
G
×
...
×
G
[cf.
the
notation
“|F
l
|”,
“F
l
”
that
appears
in
[IUTchII],
Corollaries
3.5,
3.6,
4.5,
4.6]
in
a
product
of
copies
of
G
that,
a
priori,
may
only
be
identified
with
one
another
up
to
conjugacy
[i.e.,
up
to
composition
with
an
inner
automorphism]
—
is
to
try
to
work,
instead,
with
the
(G
×
.
.
.
×
G)-conjugacy
class
of
such
a
diagonal.
Here,
to
simplify
the
notation,
let
us
assume
that
the
above
products
of
copies
of
G
are,
in
fact,
products
of
two
copies
of
G.
Then
to
identify
the
diagonal
embedding
G
→
G
×
G
with
its
(G
×
G)-conjugates
implies
that
one
must
consider
identifications
(g,
g)
∼
(g,
hgh
−1
)
=
(g,
[h,
g]
·
g)
[where
g,
h
∈
G]
—
i.e.,
one
must
identify
(g,
g)
with
the
product
of
(g,
g)
with
(1,
[h,
g]).
On
the
other
hand,
the
original
purpose
of
working
with
distinct
copies
of
G
lies
in
considering
distinct
Galois-theoretic
Kummer
classes
—
corre-
sponding
to
distinct
theta
values
[cf.
[IUTchII],
Corollaries
3.5,
3.6]
—
at
distinct
components.
That
is
to
say,
to
identify
elements
of
G
×
G
that
differ
by
a
factor
of
(1,
[h,
g])
is
incompatible,
in
an
essential
way,
with
the
convention
that
such
a
factor
(1,
[h,
g])
should
correspond
to
distinct
elements
[i.e.,
“1”
and
“[h,
g]”]
at
distinct
components
[cf.
the
discussion
of
Remark
1.5.3,
(ii),
below].
Here,
we
note
that
this
incompatibility
may
be
thought
of
as
an
essential
consequence
of
the
highly
nonabelian
nature
of
G,
e.g.,
when
G
is
taken
to
be
a
copy
of
G
v
,
for
v
∈
V
non
.
Thus,
in
summary,
this
naive
approach
to
replacing
the
theory
of
conju-
gate
synchronization
by
a
simpler
apparatus
is
inadequate
from
the
point
of
view
of
the
theory
of
the
present
series
of
papers.
(iii)
At
a
purely
combinatorial
level,
the
notion
of
conjugate
synchronization
is
reminiscent
of
the
label
synchronization
discussed
in
[IUTchI],
Remark
4.9.2,
(i),
(ii).
Indeed,
both
conjugate
and
label
synchronization
may
be
thought
of
as
a
sort
of
combinatorial
representation
of
the
arithmetic
holomorphic
structure
associated
to
a
single
vertical
line
of
the
log-theta-lattice
[cf.
the
discussion
of
[IUTchI],
Remark
4.9.2,
(iv)].
Remark
1.5.2.
(i)
Recall
that
unlike
the
case
with
the
action
of
the
F
±
l
-symmetry
on
the
various
labeled
copies
of
the
absolute
Galois
group
G
v
,
for
v
∈
V
non
[cf.
[IUTchII],
Corollaries
4.5,
(iii);
4.6,
(iii)],
it
is
not
possible
to
establish
an
analogous
theory
of
conjugate
synchronization
in
the
case
of
the
F
l
-symmetry
for
labeled
copies
of
F
[cf.
[IUTchII],
Remark
4.7.2].
This
is
to
say,
the
closest
analogue
of
the
conjugate
synchronization
obtained
in
the
local
case
relative
to
the
F
±
l
-symmetry
is
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
53
action
of
the
F
l
-symmetry
on
labeled
copies
of
the
subfields
F
mod
⊆
F
sol
⊆
F
and
the
pseudo-monoid
of
∞
κ-coric
rational
functions,
i.e.,
as
discussed
in
[IUTchII],
Corollaries
4.7,
(ii);
4.8,
(ii).
One
consequence
of
this
incompatibility
of
the
F
l
-
symmetry
with
the
full
algebraic
closure
F
of
F
mod
is
that,
as
discussed
in
[IUTchI],
Remark
5.1.5,
the
reconstruction
of
the
ring
structure
on
labeled
copies
of
the
subfield
F
sol
⊆
F
subject
to
the
F
l
-symmetry
[cf.
[IUTchII],
Corollaries
4.7,
(ii);
4.8,
(ii)],
fails
to
be
compatible
with
the
various
localization
operations
that
occur
in
the
structure
of
a
D-ΘNF-Hodge
theater.
This
is
one
quite
essential
×
”
reason
why
it
is
not
possible
to
establish
bi-coricity
properties
for,
say,
“F
sol
×
[which
we
regard
as
being
equipped
with
the
ring
structure
on
the
union
of
“F
sol
”
×
with
{0}
—
without
which
the
abstract
pair
“Gal(F
sol
/F
mod
)
F
sol
”
consisting
of
an
abstract
module
equipped
with
the
action
of
an
abstract
topological
group
is
not
very
interesting]
that
are
analogous
to
the
bi-coricity
properties
established
in
Theorem
1.5,
(iii),
for
“O
×μ
”
[cf.
the
discussion
of
Remark
1.5.1,
(i)].
From
this
point
of
view,
the
bi-coric
mono-analytic
global
realified
Frobenioids
of
Theorem
1.5,
(v)
—
i.e.,
in
essence,
the
notion
of
“log-volume”
[cf.
the
point
of
view
of
Remark
1.2.2,
(v)]
—
may
be
thought
of
as
a
sort
of
“closest
×
”
[i.e.,
which
does
not
possible
approximation”
to
such
a
“bi-coric
F
sol
exist].
Alternatively,
from
the
point
of
view
of
the
theory
to
be
developed
in
§3
below,
we
shall
apply
the
bi-coric
“O
×μ
’s”
of
Theorem
1.5,
(iii)
—
i.e.,
in
the
form
of
the
bi-coric
mono-analytic
log-shells
of
Theorem
1.5,
(iv)
—
to
construct
“multiradial
containers”
for
the
labeled
copies
of
F
mod
discussed
above
by
applying
the
localization
functors
discussed
in
[IUTchII],
Corollaries
4.7,
(iii);
4.8,
(iii).
That
is
to
say,
such
“multiradial
containers”
will
play
the
role
of
a
transportation
×
”
—
up
to
certain
indeterminacies!
—
between
distinct
arith-
mechanism
for
“F
mod
metic
holomorphic
structures
[i.e.,
distinct
vertical
lines
of
the
log-theta-lattice].
(ii)
In
the
context
of
the
discussion
of
“multiradial
containers”
in
(i)
above,
we
recall
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.2,
(ii)]
that,
in
general,
Kummer
theory
plays
a
crucial
role
precisely
in
situations
in
which
one
performs
construc-
tions
—
such
as,
for
instance,
the
construction
of
the
Θ-,
Θ
×μ
-,
or
Θ
×μ
gau
-links
—
that
are
“not
bound
to
conventional
scheme
theory”.
That
is
to
say,
in
the
case
of
the
labeled
copies
of
“F
mod
”
discussed
in
(i),
the
incompatibility
of
“solv-
able
reconstructions”
of
the
ring
structure
with
the
localization
operations
that
occur
in
a
D-ΘNF-Hodge
theater
[cf.
[IUTchI],
Remark
5.1.5]
may
be
thought
of
as
a
reflection
of
the
dismantling
of
the
global
prime-tree
structure
of
a
number
field
[cf.
the
discussion
of
[IUTchII],
Remark
4.11.2,
(iv)]
that
underlies
the
construction
of
the
Θ
±ell
NF-Hodge
theater
performed
in
[IUTchI],
[IUTchII],
hence,
in
particular,
as
a
reflection
of
the
requirement
of
establishing
a
Kummer-
compatible
theory
of
conjugate
synchronization
relative
to
the
F
±
l
-symmetry
[cf.
the
discussion
of
Remark
1.5.1,
(i)].
×
”
to
admit
a
natural
bi-coric
(iii)
Despite
the
failure
of
labeled
copies
of
“F
mod
structure
—
a
state
of
affairs
that
forces
one
to
resort
to
the
use
of
“multiradial
54
SHINICHI
MOCHIZUKI
×
containers”
in
order
to
transport
such
labeled
copies
of
“F
mod
”
to
alien
arithmetic
holomorphic
structures
[cf.
the
discussion
of
(i)
above]
—
the
global
Frobenioids
×
associated
to
copies
of
“F
mod
”
nevertheless
possess
important
properties
that
are
not
satisfied,
for
instance,
by
the
bi-coric
global
realified
Frobenioids
discussed
in
Theorem
1.5,
(v)
[cf.
also
[IUTchI],
Definition
5.2,
(iv);
[IUTchII],
Corollary
4.5,
(ii);
[IUTchII],
Corollary
4.6,
(ii)].
Indeed,
unlike
the
objects
contained
in
the
realified
global
Frobenioids
that
appear
in
Theorem
1.5,
(v),
the
objects
contained
×
”
correspond
to
genuine
in
the
global
Frobenioids
associated
to
copies
of
“F
mod
“conventional
arithmetic
line
bundles”.
In
particular,
by
applying
the
ring
structure
of
the
copies
of
“F
mod
”
under
consideration,
one
can
push
forward
such
arithmetic
line
bundles
so
as
to
obtain
arithmetic
vector
bundles
over
[the
ring
of
rational
integers]
Z
and
then
form
tensor
products
of
such
arithmetic
vector
bundles.
Such
operations
will
play
a
key
role
in
the
theory
of
§3
below,
as
well
as
in
the
theory
to
be
developed
in
[IUTchIV].
Remark
1.5.3.
(i)
In
[QuCnf]
[cf.
also
[AbsTopIII],
Proposition
2.6;
[AbsTopIII],
Corollary
2.7],
a
theory
was
developed
concerning
deformations
of
holomorphic
structures
on
Riemann
surfaces
in
which
holomorphic
structures
are
represented
by
means
of
squares
or
rectangles
on
the
surface,
while
quasiconformal
Teichmüller
deforma-
tions
of
holomorphic
structures
are
represented
by
parallelograms
on
the
surface.
That
is
to
say,
relative
to
suitable
choices
of
local
coordinates,
quasiconformal
Te-
ichmüller
deformations
may
be
thought
of
as
affine
linear
deformations
in
which
one
of
the
two
underlying
real
dimensions
of
the
Riemann
surface
is
dilated
by
some
factor
∈
R
>0
,
while
the
other
underlying
real
dimensions
is
left
undeformed.
From
this
point
of
view,
the
theory
of
conjugate
synchronization
—
which
may
be
regarded
as
a
sort
of
rigidity
that
represents
the
arithmetic
holomorphic
struc-
ture
associated
to
a
vertical
line
of
the
log-theta-lattice
[cf.
the
discussion
given
in
[IUTchII],
Remarks
4.7.3,
4.7.4,
of
the
uniradiality
of
the
F
±
l
-symmetry
that
underlies
the
phenomenon
of
conjugate
synchronization]
—
may
be
thought
of
as
a
sort
of
nonarchimedean
arithmetic
analogue
of
the
representation
of
holo-
morphic
structures
by
means
of
squares/rectangles
referred
to
above.
That
is
to
say,
the
right
angles
which
are
characteristic
of
squares/rectangles
may
be
thought
of
as
a
sort
of
synchronization
between
the
metrics
of
the
two
underlying
real
di-
mensions
of
a
Riemann
surface
[i.e.,
metrics
which,
a
priori,
may
differ
by
some
dilating
factor]
—
cf.
Fig.
1.4
below.
Here,
we
mention
in
passing
that
this
point
of
view
is
reminiscent
of
the
discussion
of
[IUTchII],
Remark
3.6.5,
(ii),
in
which
the
point
of
view
is
taken
that
the
phenomenon
of
conjugate
synchronization
may
be
thought
of
as
a
reflection
of
the
coherence
of
the
arithmetic
holomorphic
structures
involved.
(ii)
Relative
to
the
point
of
view
discussed
in
(i),
the
approach
described
in
Re-
mark
1.5.1,
(ii),
to
“avoiding
conjugate
synchronization
by
identifying
the
various
conjugates
of
the
diagonal
embedding”
corresponds
—
in
light
of
the
highly
non-
abelian
nature
of
the
groups
involved!
[cf.
the
discussion
of
Remark
1.5.1,
(ii)]
—
to
thinking
of
a
holomorphic
structure
on
a
Riemann
surface
as
an
“equivalence
class
of
holomorphic
structures
in
the
usual
sense
relative
to
the
equivalence
relation
of
differing
by
a
Teichmüller
deformation”!
That
is
to
say,
such
an
[unconventional!]
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
55
approach
to
the
definition
of
a
holomorphic
structure
allows
one
to
circumvent
the
issue
of
rigidifying
the
relationship
between
the
metrics
of
the
two
underlying
real
dimensions
of
the
Riemann
surface
—
but
only
at
the
cost
of
rendering
unfeasible
any
meaningful
theory
of
“deformations
of
a
holomorphic
structure”!
(iii)
The
analogy
discussed
in
(i)
between
conjugate
synchronization
[which
arises
from
the
F
±
l
-symmetry!]
and
the
representation
of
a
complex
holomor-
phic
structure
by
means
of
squares/rectangles
may
also
be
applied
to
the
“κ-sol-
conjugate
synchronization”
[cf.
the
discussion
of
[IUTchI],
Remark
5.1.5]
given
in
[IUTchII],
Corollary
4.7,
(ii);
[IUTchII],
Corollary
4.8,
(ii),
between,
for
in-
stance,
the
various
labeled
non-realified
and
realified
global
Frobenioids
by
means
of
the
F
l
-symmetry.
Indeed,
this
analogy
is
all
the
more
apparent
in
the
case
of
the
realified
global
Frobenioids
—
which
admit
a
natural
R
>0
-action.
Here,
we
observe
in
passing
that,
just
as
the
theory
of
conjugate
synchronization
[via
the
F
±
l
-symmetry]
plays
an
essential
role
in
the
construction
of
the
local
portions
of
the
Θ
×μ
-,
Θ
×μ
gau
-links
given
in
[IUTchII],
Corollary
4.10,
(i),
(ii),
(iii),
the
synchronization
of
global
realified
Frobenioids
by
means
of
the
F
l
-symmetry
may
be
related
—
via
the
isomorphisms
of
Frobenioids
of
the
second
displays
of
[IUTchII],
Corollary
4.7,
(iii);
[IUTchII],
Corollary
4.8,
(iii)
[cf.
also
the
discussion
of
[IUTchII],
Remark
4.8.1]
—
to
the
construction
of
the
global
realified
Frobenioid
portion
of
the
Θ
×μ
gau
-link
given
in
[IUTchII],
Corollary
4.10,
(ii).
On
the
other
hand,
the
synchronization
involving
the
non-realified
global
Frobe-
nioids
may
be
thought
of
as
a
sort
of
further
rigidification
of
the
global
realified
Frobenioids.
As
discussed
in
Remark
1.5.2,
(iii),
this
“further
rigidification”
will
play
an
important
role
in
the
theory
of
§3
below.
G
v
...
G
v
G
v
G
v
...
G
v
...
...
−l
...
−1
0
1
...
l
R
>0
..
.
..
.
..
.
...
..
.
..
.
...
...
...
...
id
Fig.
1.4:
Analogy
between
conjugate
synchronization
and
the
representation
of
complex
holomorphic
structures
via
squares/rectangles
56
SHINICHI
MOCHIZUKI
Remark
1.5.4.
(i)
As
discussed
in
[IUTchII],
Remark
3.8.3,
(iii),
one
of
the
main
themes
of
the
present
series
of
papers
is
the
goal
of
giving
an
explicit
description
of
what
one
arithmetic
holomorphic
structure
—
i.e.,
one
vertical
line
of
the
log-theta-lattice
—
looks
like
from
the
point
of
view
of
a
distinct
arithmetic
holomorphic
structure
—
i.e.,
another
vertical
line
of
the
log-theta-lattice
—
that
is
only
related
to
the
original
arithmetic
holomorphic
structure
via
some
mono-analytic
core,
e.g.,
the
various
bi-coric
structures
discussed
in
Theorem
1.5,
(iii),
(iv),
(v).
Typically,
the
objects
of
interest
that
are
constructed
within
the
original
arithmetic
holomorphic
structure
are
Frobenius-like
structures
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.2],
which,
as
we
recall
from
the
discussion
of
Remark
1.5.2,
(ii)
[cf.
also
the
discussion
of
[IUTchII],
Remark
3.6.2,
(ii)],
are
necessary
in
order
to
perform
con-
structions
—
such
as,
for
instance,
the
construction
of
the
Θ-,
Θ
×μ
-,
or
Θ
×μ
gau
-links
—
that
are
“not
bound
to
conventional
scheme
theory”.
Indeed,
the
main
example
of
such
an
object
of
interest
consists
precisely
of
the
Gaussian
monoids
discussed
in
[IUTchII],
§3,
§4.
Thus,
the
operation
of
describing
such
an
object
of
interest
from
the
point
of
view
of
a
distinct
arithmetic
holomorphic
structure
may
be
broken
down
into
two
steps:
(a)
passing
from
Frobenius-like
structures
to
étale-like
structures
via
various
Kummer
isomorphisms;
(b)
transporting
the
resulting
étale-like
structures
from
one
arithmetic
holo-
morphic
structure
to
another
by
means
of
various
multiradiality
prop-
erties.
In
particular,
the
computation
of
what
the
object
of
interest
looks
like
from
the
point
of
view
of
a
distinct
arithmetic
holomorphic
structure
may
be
broken
down
into
the
computation
of
the
indeterminacies
or
“departures
from
rigidity”
that
arise
—
i.e.,
the
computation
of
“what
sort
of
damage
is
incurred
to
the
object
of
interest”
—
during
the
execution
of
each
of
these
two
steps
(a),
(b).
We
shall
refer
to
the
indeterminacies
that
arise
from
(a)
as
Kummer-detachment
inde-
terminacies
and
to
the
indeterminacies
that
arise
from
(b)
as
étale-transport
indeterminacies.
(ii)
Étale-transport
indeterminacies
typically
amount
to
the
indeterminacies
that
occur
as
a
result
of
the
execution
of
various
“anabelian”
or
“group-theoretic”
algorithms.
One
fundamental
example
of
such
indeterminacies
is
constituted
by
the
indeterminacies
that
occur
in
the
context
of
Theorem
1.5,
(iii),
(iv),
as
a
result
of
the
existence
of
automorphisms
of
the
various
[copies
of]
local
absolute
Galois
groups
G
v
,
for
v
∈
V
non
,
which
are
not
of
scheme-theoretic
origin
[cf.
the
discussion
of
[AbsTopIII],
§I3].
(iii)
On
the
other
hand,
one
important
example,
from
the
point
of
view
of
the
theory
of
the
present
series
of
papers,
of
a
Kummer-detachment
indeterminacy
is
constituted
by
the
Frobenius-picture
diagrams
given
in
Propositions
1.2,
(x);
1.3,
(iv)
—
i.e.,
the
issue
of
which
path
one
is
to
take
from
a
particular
“•”
to
the
coric
“◦”.
That
is
to
say,
despite
the
fact
that
these
diagrams
fail
to
be
commutative,
the
“upper
semi-commutativity”
property
satisfied
by
the
coric
holomorphic
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
57
log-shells
involved
[cf.
the
discussion
of
Remark
1.2.2,
(iii)]
may
be
regarded
as
a
sort
of
computation,
in
the
form
of
an
upper
estimate,
of
the
Kummer-detachment
indeterminacy
in
question.
Another
important
example,
from
the
point
of
view
of
the
theory
of
the
present
series
of
papers,
of
a
Kummer-detachment
indeterminacy
is
×
-indeterminacies
discussed
in
Remark
1.4.2
[cf.
also
the
Kummer
given
by
the
Z
isomorphisms
of
the
final
display
of
Theorem
1.5,
(iii)].
58
SHINICHI
MOCHIZUKI
Section
2:
Multiradial
Theta
Monoids
In
the
present
§2,
we
globalize
the
multiradial
portion
of
the
local
the-
ory
of
theta
monoids
developed
in
[IUTchII],
§1,
§3,
at
v
∈
V
bad
[cf.,
espe-
cially,
[IUTchII],
Corollary
1.12;
[IUTchII],
Proposition
3.4]
so
as
to
cover
the
theta
monoids/Frobenioids
of
[IUTchII],
Corollaries
4.5,
(iv),
(v);
4.6,
(iv),
(v),
and
ex-
plain
how
the
resulting
theory
may
be
fit
into
the
framework
of
the
log-theta-
lattice
developed
in
§1.
In
the
following
discussion,
we
assume
that
we
have
been
given
initial
Θ-data
±ell
as
in
[IUTchI],
Definition
3.1.
Let
†
HT
Θ
NF
be
a
Θ
±ell
NF-Hodge
theater
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
Definition
6.13,
(i)]
and
{
n,m
HT
Θ
±ell
NF
}
n,m∈Z
a
collection
of
distinct
Θ
±ell
NF-Hodge
theaters
[relative
to
the
given
initial
Θ-data]
indexed
by
pairs
of
integers,
which
we
think
of
as
arising
from
a
Gaussian
log-theta-
lattice,
as
in
Definition
1.4.
We
begin
by
reviewing
the
theory
of
theta
monoids
developed
in
[IUTchII].
Proposition
2.1.
(Vertical
Coricity
and
Kummer
Theory
of
Theta
Monoids)
We
maintain
the
notation
introduced
above.
Also,
we
shall
use
the
notation
Aut
F
(−)
to
denote
the
group
of
automorphisms
of
the
F
-prime-strip
in
parentheses.
Then:
(i)
(Vertically
Coric
Theta
Monoids)
In
the
notation
of
[IUTchII],
Corol-
lary
4.5,
(iv),
(v)
[cf.
also
the
assignment
“0,
→
>”
of
[IUTchI],
Proposition
6.7],
there
are
functorial
algorithms
in
the
D-
and
D
-prime-strips
†
D
>
,
†
D
>
±ell
associated
to
the
Θ
±ell
NF-Hodge
theater
†
HT
Θ
NF
for
constructing
collections
of
data
indexed
by
V
V
v
→
Ψ
env
(
†
D
>
)
v
;
V
v
→
∞
Ψ
env
(
†
D
>
)
v
as
well
as
a
global
realified
Frobenioid
†
D
env
(
D
>
)
∼
†
equipped
with
a
bijection
Prime(D
env
(
D
>
))
→
V
and
corresponding
local
isomor-
phisms,
for
each
v
∈
V,
as
described
in
detail
in
[IUTchII],
Corollary
4.5,
(v).
In
particular,
each
isomorphism
of
the
full
poly-isomorphism
induced
[cf.
Theorem
1.5,
(i)]
by
a
vertical
arrow
of
the
Gaussian
log-theta-lattice
under
consideration
induces
a
compatible
collection
of
isomorphisms
∼
Ψ
env
(
n,m
D
>
)
→
Ψ
env
(
n,m+1
D
>
);
∞
Ψ
env
(
∼
n,m
∼
D
>
)
→
n,m
n,m+1
(
D
>
)
→
D
env
(
D
>
)
D
env
∞
Ψ
env
(
n,m+1
D
>
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
59
—
where
the
final
isomorphism
of
Frobenioids
is
compatible
with
the
respective
bijections
involving
“Prime(−)”,
as
well
as
with
the
respective
local
isomorphisms
for
each
v
∈
V.
(ii)
(Kummer
Isomorphisms)
In
the
notation
of
[IUTchII],
Corollary
4.6,
±ell
(iv),
(v),
there
are
functorial
algorithms
in
the
Θ
±ell
NF-Hodge
theater
†
HT
Θ
NF
for
constructing
collections
of
data
indexed
by
V
V
v
→
Ψ
F
env
(
†
HT
Θ
)
v
;
V
v
→
∞
Ψ
F
env
(
†
HT
Θ
)
v
as
well
as
a
global
realified
Frobenioid
†
C
env
(
HT
Θ
)
∼
†
equipped
with
a
bijection
Prime(C
env
(
HT
Θ
))
→
V
and
corresponding
local
iso-
morphisms,
for
each
v
∈
V,
as
described
in
detail
in
[IUTchII],
Corollary
4.6,
±ell
(v).
Moreover,
there
are
functorial
algorithms
in
†
HT
Θ
NF
for
constructing
Kummer
isomorphisms
∼
Ψ
F
env
(
†
HT
Θ
)
→
Ψ
env
(
†
D
>
);
†
C
env
(
HT
Θ
)
Θ
†
∞
Ψ
F
env
(
HT
)
∼
→
∼
→
†
∞
Ψ
env
(
D
>
)
†
D
env
(
D
>
)
—
where
the
final
isomorphism
of
Frobenioids
is
compatible
with
the
respective
bi-
jections
involving
“Prime(−)”,
as
well
as
with
the
respective
local
isomorphisms
for
each
v
∈
V
—
with
the
data
discussed
in
(i)
[cf.
[IUTchII],
Corollary
4.6,
(iv),
(v)].
Finally,
the
collection
of
data
Ψ
env
(
†
D
>
)
gives
rise,
in
a
natural
fash-
ion,
to
an
F
-prime-strip
F
env
(
†
D
>
)
[cf.
the
F
-prime-strip
“
†
F
env
”
of
[IUTchII],
†
(
D
>
),
equipped
with
the
Corollary
4.10,
(ii)];
the
global
realified
Frobenioid
D
env
∼
†
bijection
Prime(D
env
(
D
>
))
→
V
and
corresponding
local
isomorphisms,
for
each
v
∈
V,
reviewed
in
(i),
together
with
the
F
-prime-strip
F
env
(
†
D
>
),
determine
an
†
†
F
-prime-strip
F
env
(
D
>
)
[cf.
the
F
-prime-strip
“
F
env
”
of
[IUTchII],
Corollary
4.10,
(ii)].
In
particular,
the
first
and
third
Kummer
isomorphisms
of
the
above
display
may
be
interpreted
as
[compatible]
isomorphisms
†
F
env
∼
→
F
env
(
†
D
>
);
†
F
env
∼
†
→
F
env
(
D
>
)
of
F
-,
F
-prime-strips.
(iii)
(Kummer
Theory
at
Bad
Primes)
The
portion
at
v
∈
V
bad
of
the
Kummer
isomorphisms
of
(ii)
is
obtained
by
composing
the
Kummer
isomorphisms
of
[IUTchII],
Proposition
3.3,
(i)
—
which,
we
recall,
were
defined
by
forming
Kummer
classes
in
the
context
of
mono-theta
environments
that
arise
from
tempered
Frobenioids
—
with
the
isomorphisms
on
cohomology
classes
induced
[cf.
the
upper
left-hand
portion
of
the
first
display
of
[IUTchII],
Proposition
3.4,
(i)]
by
the
full
poly-isomorphism
of
projective
systems
of
mono-theta
envi-
∼
†
Θ
†
ronments
“M
Θ
∗
(
D
>,v
)
→
M
∗
(
F
v
)”
[cf.
[IUTchII],
Proposition
3.4;
[IUTchII],
Remark
4.2.1,
(iv)]
between
projective
systems
of
mono-theta
environments
that
arise
from
tempered
Frobenioids
[i.e.,
“
†
F
v
”]
and
projective
systems
of
mono-theta
60
SHINICHI
MOCHIZUKI
environments
that
arise
from
the
tempered
fundamental
group
[i.e.,
“
†
D
>,v
”]
—
cf.
the
left-hand
portion
of
the
third
display
of
[IUTchII],
Corollary
3.6,
(ii),
in
the
context
of
the
discussion
of
[IUTchII],
Remark
3.6.2,
(i).
Here,
each
“isomorphism
on
cohomology
classes”
is
induced
by
the
isomorphism
on
exterior
cyclotomes
†
Π
μ
(M
Θ
∗
(
D
>,v
))
∼
→
†
Π
μ
(M
Θ
∗
(
F
v
))
determined
by
each
of
the
isomorphisms
that
constitutes
the
full
poly-isomorphism
of
projective
systems
of
mono-theta
environments
discussed
above.
In
particular,
the
composite
map
†
Π
μ
(M
Θ
∗
(
D
>,v
))
⊗
Q/Z
(Ψ
†
F
v
Θ
)
×μ
→
obtained
by
composing
the
result
of
applying
“⊗
Q/Z”
to
this
isomorphism
on
ex-
terior
cyclotomes
with
the
natural
inclusion
†
Π
μ
(M
Θ
∗
(
F
v
))
⊗
Q/Z
(Ψ
†
F
v
Θ
)
×
→
[cf.
the
notation
of
[IUTchII],
Proposition
3.4,
(i);
the
description
given
in
[IUTchII],
Proposition
1.3,
(i),
of
the
exterior
cyclotome
of
a
mono-theta
environment
that
arises
from
a
tempered
Frobenioid]
and
the
natural
projection
(Ψ
†
F
v
Θ
)
×
(Ψ
†
F
v
Θ
)
×μ
is
equal
to
the
zero
map.
(iv)
(Kummer
Theory
at
Good
Nonarchimedean
Primes)
The
unit
portion
at
v
∈
V
good
V
non
of
the
Kummer
isomorphisms
of
(ii)
is
obtained
[cf.
[IUTchII],
Proposition
4.2,
(iv)]
as
the
unit
portion
of
a
“labeled
version”
of
the
isomorphism
of
ind-topological
monoids
equipped
with
a
topological
group
action
—
i.e.,
in
the
language
of
[AbsTopIII],
Definition
3.1,
(ii),
the
isomorphism
of
“MLF-Galois
TM-pairs”
—
discussed
in
[IUTchII],
Proposition
4.2,
(i)
[cf.
also
[IUTchII],
Remark
1.11.1,
(i),
(a);
[AbsTopIII],
Proposition
3.2,
(iv)].
In
particular,
the
portion
at
v
∈
V
good
V
non
of
the
Aut
F
(
†
F
env
)-orbit
of
the
second
isomorphism
of
the
final
display
of
(ii)
may
be
obtained
as
a
“labeled
version”
of
the
“Kummer
poly-isomorphism
of
semi-simplifications”
given
in
the
final
display
of
[IUTchII],
Proposition
4.2,
(ii).
(v)
(Kummer
Theory
at
Archimedean
Primes)
The
unit
portion
at
v
∈
V
arc
of
the
Kummer
isomorphisms
of
(ii)
is
obtained
[cf.
[IUTchII],
Propo-
sition
4.4,
(iv)]
as
the
unit
portion
of
a
“labeled
version”
of
the
isomorphism
of
topological
monoids
discussed
in
[IUTchII],
Proposition
4.4,
(i).
In
particular,
the
portion
at
v
∈
V
arc
of
the
Aut
F
(
†
F
env
)-orbit
of
the
second
isomorphism
of
the
final
display
of
(ii)
may
be
obtained
as
a
“labeled
version”
of
the
“Kummer
poly-
isomorphism
of
semi-simplifications”
given
in
the
final
display
of
[IUTchII],
Proposition
4.4,
(ii)
[cf.
also
[IUTchII],
Remark
4.6.1].
(vi)
(Compatibility
with
Constant
Monoids)
The
definition
of
the
unit
portion
of
the
theta
monoids
involved
[cf.
[IUTchII],
Corollary
4.10,
(iv)]
gives
rise
to
natural
isomorphisms
†
×
F
∼
→
†
×
F
env
;
∼
†
F
×
(
†
D
)
→
F
×
env
(
D
>
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
61
—
i.e.,
where
the
morphism
induced
on
F
×μ
-prime-strips
by
the
first
displayed
isomorphism
is
precisely
the
isomorphism
of
the
first
display
of
[IUTchII],
Corollary
4.10,
(iv)
—
of
the
respective
associated
F
×
-prime-strips
[cf.
the
notation
of
Theorem
1.5,
(iii),
where
the
label
“n,
m”
is
replaced
by
the
label
“†”].
Moreover,
these
natural
isomorphisms
are
compatible
with
the
Kummer
isomorphisms
of
(ii)
above
and
Theorem
1.5,
(iii).
Proof.
The
various
assertions
of
Proposition
2.1
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
2.1.1.
The
theory
of
mono-theta
environments
[cf.
Proposition
2.1,
(iii)]
will
play
a
crucial
role
in
the
theory
of
the
present
§2
[cf.
Theorem
2.2,
(ii);
Corollary
2.3,
(iv),
below]
in
the
passage
from
Frobenius-like
to
étale-like
struc-
tures
[cf.
Remark
1.5.4,
(i),
(a)]
at
bad
primes.
In
particular,
the
various
rigidity
properties
of
mono-theta
environments
established
in
[EtTh]
play
a
fundamental
role
in
ensuring
that
the
resulting
“Kummer-detachment
indeterminacies”
[cf.
the
discussion
of
Remark
1.5.4,
(i)]
are
sufficiently
mild
so
as
to
allow
the
establishment
of
the
various
reconstruction
algorithms
of
interest.
For
this
reason,
we
pause
to
review
the
main
properties
of
mono-theta
environments
established
in
[EtTh]
[cf.
[EtTh],
Introduction]
—
namely,
(a)
cyclotomic
rigidity
(b)
discrete
rigidity
(c)
constant
multiple
rigidity
(d)
isomorphism
class
compatibility
(e)
Frobenioid
structure
compatibility
—
and
the
roles
played
by
these
main
properties
in
the
theory
of
the
present
series
of
papers.
Here,
we
remark
that
“isomorphism
class
compatibility”
[i.e.,
(d)]
refers
to
compatibility
with
the
convention
that
various
objects
of
the
tempered
Frobenioids
[and
their
associated
base
categories]
under
consideration
are
known
only
up
to
isomorphism
[cf.
[EtTh],
Corollary
5.12;
[EtTh],
Remarks
5.12.1,
5.12.2].
In
the
Introduction
to
[EtTh],
instead
of
referring
to
(d)
in
this
form,
we
referred
to
the
property
of
compatibility
with
the
topology
of
the
tempered
fundamental
group.
In
fact,
however,
this
compatibility
with
the
topology
of
the
tempered
fundamental
group
is
a
consequence
of
(d)
[cf.
[EtTh],
Remarks
5.12.1,
5.12.2].
On
the
other
hand,
from
the
point
of
view
of
the
present
series
of
papers,
the
essential
property
of
interest
in
this
context
is
best
understood
as
being
the
property
(d).
(i)
First,
we
recall
that
the
significance,
in
the
context
of
the
theory
of
the
present
series
of
papers,
of
the
compatibility
with
the
Frobenioid
structure
of
the
tempered
Frobenioids
under
consideration
[i.e.,
(e)]
—
i.e.,
in
particular,
with
the
monoidal
portion,
equipped
with
its
natural
Galois
action,
of
these
Frobenioids
—
lies
in
the
role
played
by
this
“Frobenius-like”
monoidal
portion
in
performing
constructions
—
such
as,
for
instance,
the
construction
of
the
log-,
Θ-,
Θ
×μ
-,
or
Θ
×μ
gau
-links
—
that
are
“not
bound
to
conventional
scheme
theory”,
but
may
be
related,
via
Kummer
theory,
to
various
étale-like
structures
[cf.
the
discussions
of
Remark
1.5.4,
(i);
[IUTchII],
Remark
3.6.2,
(ii);
[IUTchII],
Remark
3.6.4,
(ii),
(v)].
62
SHINICHI
MOCHIZUKI
(ii)
Next,
we
consider
isomorphism
class
compatibility
[i.e.,
(d)].
As
discussed
above,
this
compatibility
corresponds
to
regarding
each
of
the
various
objects
of
the
tempered
Frobenioids
[and
their
associated
base
categories]
under
consideration
as
being
known
only
up
to
isomorphism
[cf.
[EtTh],
Corollary
5.12;
[EtTh],
Re-
marks
5.12.1,
5.12.2].
As
discussed
in
[IUTchII],
Remark
3.6.4,
(i),
the
significance
of
this
property
(d)
in
the
context
of
the
present
series
of
papers
lies
in
the
fact
that
—
unlike
the
case
with
the
projective
systems
constituted
by
Kummer
tow-
ers
constructed
from
N
-th
power
morphisms,
which
are
compatible
with
only
the
multiplicative,
but
not
the
additive
structures
of
the
p
v
-adic
local
fields
involved
—
each
individual
object
in
such
a
Kummer
tower
corresponds
to
a
single
field
[i.e.,
as
opposed
to
a
projective
system
of
multiplicative
groups
of
fields].
This
field/ring
structure
is
necessary
in
order
to
apply
the
theory
of
the
log-link
developed
in
§1
—
cf.
the
vertical
coricity
discussed
in
Proposition
2.1,
(i).
Note,
moreover,
that,
unlike
the
log-,
Θ-,
Θ
×μ
-,
or
Θ
×μ
gau
-links,
the
N
-th
power
morphisms
that
appear
in
a
Kummer
tower
are
“algebraic”,
hence
compatible
with
the
conven-
tional
scheme
theory
surrounding
the
étale
[or
tempered]
fundamental
group.
In
particular,
since
the
tempered
Frobenioids
under
consideration
may
be
constructed
from
such
scheme-theoretic
categories,
the
fundamental
groups
on
either
side
of
such
an
N
-th
power
morphism
may
be
related
up
to
an
indeterminacy
arising
from
an
inner
automorphism
of
the
tempered
fundamental
group
[i.e.,
the
“funda-
mental
group”
of
the
base
category]
under
consideration
—
cf.
the
discussion
of
[IUTchII],
Remark
3.6.3,
(ii).
On
the
other
hand,
the
objects
that
appear
in
these
Kummer
towers
necessarily
arise
from
nontrivial
line
bundles
[indeed,
line
bundles
all
of
whose
positive
tensor
powers
are
nontrivial!]
on
tempered
coverings
of
a
Tate
curve
—
cf.
the
constructions
underlying
the
Frobenioid-theoretic
version
of
the
mono-theta
environment
[cf.
[EtTh],
Proposition
1.1;
[EtTh],
Lemma
5.9];
the
crucial
role
played
by
the
commutator
“[−,
−]”
in
the
theory
of
cyclotomic
rigidity
[i.e.,
(a)]
reviewed
in
(iv)
below.
In
particular,
the
extraction
of
various
N
-th
roots
in
a
Kummer
tower
necessarily
leads
to
mutually
non-isomorphic
line
bundles,
i.e.,
mutually
non-isomorphic
objects
in
the
Kummer
tower.
From
the
point
of
view
of
reconstruction
algorithms,
such
non-isomorphic
objects
may
be
naturally
—
i.e.,
algorithmically
—
related
to
another
only
via
indeterminate
isomorphisms
[cf.
(d)!].
This
point
of
view
is
precisely
the
starting
point
of
the
discussion
of
—
for
instance,
“constant
multiple
indeterminacy”
in
—
[EtTh],
Remarks
5.12.2,
5.12.3.
(iii)
Next,
we
recall
that
the
significance
of
constant
multiple
rigidity
[i.e.,
(c)]
in
the
context
of
the
present
series
of
papers
lies
in
the
construction
of
the
canonical
splittings
of
theta
monoids
via
restriction
to
the
zero
section
discussed,
for
instance,
in
[IUTchII],
Corollary
1.12,
(ii);
[IUTchII],
Proposition
3.3,
(i);
[IUTchII],
Remark
1.12.2,
(iv)
[cf.
also
Remark
1.2.3,
(i),
of
the
present
paper].
(iv)
Next,
we
review
the
significance
of
cyclotomic
rigidity
[i.e.,
(a)]
in
the
context
of
the
present
series
of
papers.
First,
we
recall
that
this
cyclotomic
rigidity
is
essentially
a
consequence
of
the
nondegenerate
nature
of
the
commutator
“[−,
−]”
of
the
theta
groups
involved
[cf.
the
discussion
of
[EtTh],
Introduction;
[EtTh],
Remark
2.19.2].
Put
another
way,
since
this
commutator
is
quadratic
in
nature,
one
may
think
of
this
nondegenerate
nature
of
the
commutator
as
a
statement
to
the
effect
that
“the
degree
of
the
commutator
is
precisely
2”.
At
a
more
concrete
level,
the
cyclotomic
rigidity
arising
from
a
mono-theta
environment
consists
of
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
63
a
certain
specific
isomorphism
between
the
interior
and
exterior
cyclotomes
[cf.
the
discussion
of
[IUTchII],
Definition
1.1,
(ii);
[IUTchII],
Remark
1.1.1].
Put
another
way,
one
may
think
of
this
cyclotomic
rigidity
isomorphism
as
a
sort
of
rigidification
of
a
certain
“projective
line
of
cyclotomes”,
i.e.,
the
projectivization
of
the
direct
sum
of
the
interior
and
exterior
cyclotomes
[cf.
the
computations
that
underlie
[EtTh],
Proposition
2.12].
In
particular,
this
rigidification
is
fundamentally
nonlinear
in
nature.
Indeed,
if
one
attempts
to
compose
it
with
an
N
-th
power
morphism,
then
one
is
obliged
to
sacrifice
constant
multiple
rigidity
[i.e.,
(c)]
—
cf.
the
discussion
of
[EtTh],
Remark
5.12.3.
That
is
to
say,
the
distinguished
nature
of
the
“first
power”
of
the
cyclotomic
rigidity
isomorphism
is
an
important
theme
in
the
theory
of
[EtTh]
[cf.
the
discussion
of
[EtTh],
Remark
5.12.5;
[IUTchII],
Remark
3.6.4,
(iii),
(iv)].
The
multiradiality
of
mono-theta-theoretic
cyclotomic
rigidity
[cf.
[IUTchII],
Corollary
1.10]
—
which
lies
in
stark
contrast
with
the
indeterminacies
that
arise
when
one
attempts
to
give
a
multiradial
formulation
[cf.
[IUTchII],
Corollary
1.11;
the
discussion
of
[IUTchII],
Remark
1.11.3]
of
the
more
classical
“MLF-Galois
pair
cyclotomic
rigidity”
arising
from
local
class
field
theory
—
will
play
a
central
role
in
the
theory
of
the
present
§2
[cf.
Theorem
2.2,
Corollary
2.3
below].
(v)
Finally,
we
review
the
significance
of
discrete
rigidity
[i.e.,
(b)]
in
the
context
of
the
present
series
of
papers.
First,
we
recall
that,
at
a
technical
level,
whereas
cyclotomic
rigidity
may
be
regarded
[cf.
the
discussion
of
(iv)]
as
a
consequence
of
the
fact
that
“the
degree
of
the
commutator
is
precisely
2”,
discrete
rigidity
may
be
regarded
as
a
consequence
of
the
fact
that
“the
degree
of
the
commutator
is
≤
2”
[cf.
the
statements
and
proofs
of
[EtTh],
Proposition
2.14,
(ii),
(iii)].
At
a
more
concrete
level,
discrete
rigidity
assures
one
that
one
may
restrict
one’s
attentions
to
Z-multiples/powers
—
as
opposed
to
Z-multiples/powers
—
of
divisors,
line
bundles,
and
rational
functions
[such
as,
for
instance,
the
q-parameter!]
on
the
tempered
coverings
of
a
Tate
curve
that
occur
in
the
theory
of
[EtTh]
[cf.
[EtTh],
Remark
2.19.4].
This
prompts
the
following
question:
Can
one
develop
a
theory
of
Z-divisors/line
bundles/rational
functions
in,
for
instance,
a
parallel
fashion
to
the
way
in
which
one
considers
perfections
and
realifications
of
Frobenioids
in
the
theory
of
[FrdI]?
As
far
as
the
author
can
see
at
the
time
of
writing,
the
answer
to
this
question
is
“no”.
Indeed,
unlike
the
case
with
Q
or
R,
there
is
no
notion
of
positivity
[or
nega-
For
instance,
−1
∈
Z
may
be
obtained
as
a
limit
of
positive
integers.
In
tivity]
in
Z.
particular,
if
one
had
a
theory
of
Z-divisors/line
bundles/rational
functions,
then
such
a
theory
would
necessarily
require
one
to
“confuse”
positive
[i.e.,
effective]
and
negative
divisors,
hence
to
work
birationally.
But
to
work
birationally
means,
in
particular,
that
one
must
sacrifice
the
conventional
structure
of
isomorphisms
[e.g.,
automorphisms]
between
line
bundles
—
which
plays
an
indispensable
role,
for
instance,
in
the
constructions
underlying
the
Frobenioid-theoretic
version
of
the
mono-theta
environment
[cf.
[EtTh],
Proposition
1.1;
[EtTh],
Lemma
5.9;
the
crucial
role
played
by
the
commutator
“[−,
−]”
in
the
theory
of
cyclotomic
rigidity
[i.e.,
(a)]
reviewed
in
(iv)
above].
64
SHINICHI
MOCHIZUKI
Remark
2.1.2.
(i)
In
the
context
of
the
discussion
of
Remark
2.1.1,
(v),
it
is
of
interest
to
recall
[cf.
[IUTchII],
Remark
4.5.3,
(iii);
[IUTchII],
Remark
4.11.2,
(iii)]
that
the
essen-
tial
role
played,
in
the
context
of
the
F
±
l
-symmetry,
by
the
“global
bookkeeping
operations”
involving
the
labels
of
the
evaluation
points
gives
rise,
in
light
of
the
profinite
nature
of
the
global
étale
fundamental
groups
involved,
to
a
situation
in
which
one
must
apply
the
“complements
on
tempered
coverings”
developed
in
[IUTchI],
§2.
That
is
to
say,
in
the
notation
of
the
discussion
given
in
[IUTchII],
Remark
2.1.1,
(i),
of
the
various
tempered
coverings
that
occur
at
v
∈
V
bad
,
these
“complements
on
tempered
coverings”
are
applied
precisely
so
as
to
allow
one
to
restrict
one’s
attention
to
the
[discrete!]
Z-conjugates
—
i.e.,
as
opposed
to
[profi-
for
the
profinite
completion
of
Z]
—
of
the
[where
we
write
Z
nite!]
Z-conjugates
theta
functions
involved.
In
particular,
although
such
“evaluation-related
issues”,
which
will
become
relevant
in
the
context
of
the
theory
of
§3
below,
do
not
play
a
role
in
the
theory
of
the
present
§2,
the
role
played
by
the
theory
of
[IUTchI],
§2,
in
the
theory
of
the
present
series
of
papers
may
also
be
thought
of
as
a
sort
of
“discrete
rigidity”
—
which
we
shall
refer
to
as
“evaluation
discrete
rigidity”
—
i.e.,
a
sort
of
rigidity
that
is
concerned
with
similar
issues
to
the
issues
discussed
in
the
case
of
“mono-theta-theoretic
discrete
rigidity”
in
Remark
2.1.1,
(v),
above.
(ii)
Next,
let
us
suppose
that
we
are
in
the
situation
discussed
in
[IUTchII],
def
for
the
profinite
completion
of
Proposition
2.1.
Fix
v
∈
V
bad
.
Write
Π
=
Π
v
;
Π
(⊆
Z).
Write
l
·
Z
Π.
Thus,
we
have
natural
surjections
Π
l
·
Z
(⊆
Z),
Π
def
Next,
we
observe
that
from
the
point
of
view
of
the
evaluation
×
Z
⊆
Π.
Π
†
=
Π
Z
points,
the
evaluation
discrete
rigidity
discussed
in
(i)
corresponds
to
the
issue
of
whether,
relative
to
some
arbitrarily
chosen
basepoint,
the
“coordinates”
[i.e.,
element
of
the
“torsor
over
Z”
discussed
in
[IUTchII],
Remark
2.1.1,
(i)]
of
the
Thus,
if
one
is
only
concerned
with
the
issue
of
evaluation
point
lie
∈
Z
or
∈
Z.
arranging
for
these
coordinates
to
lie
∈
Z,
then
one
is
led
to
pose
the
following
question:
Is
it
possible
to
simply
use
the
“partially
tempered
fundamental
group”
Π
†
instead
of
the
“full”
tempered
fundamental
group
Π
in
the
theory
of
the
present
series
of
papers?
The
answer
to
this
question
is
“no”.
One
way
to
see
this
is
to
consider
the
[easily
verified]
natural
isomorphism
∼
†
†
N
Π
(Π
)/Π
→
Z/Z
†
†
involving
the
normalizer
N
Π
(Π
)
of
Π
in
Π.
One
consequence
of
this
isomorphism
is
that
—
unlike
the
tempered
fundamental
group
Π
[cf.,
e.g.,
[SemiAnbd],
The-
orems
6.6,
6.8]
—
the
topological
group
Π
†
fails
to
satisfy
various
fundamental
absolute
anabelian
properties
which
play
a
crucial
role
in
the
theory
of
[EtTh],
as
well
as
in
the
present
series
of
papers
[cf.,
e.g.,
the
theory
of
[IUTchII],
§2].
At
a
more
concrete
level,
unlike
the
case
with
the
tempered
fundamental
group
Π,
the
profinite
conjugacy
indeterminacies
that
act
on
Π
†
give
rise
to
Z-translation
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
65
indeterminacies
acting
on
the
coordinates
of
the
evaluation
points
involved.
That
indeterminacies
are
avoided
precisely
is
to
say,
in
the
case
of
Π,
such
Z-translation
by
applying
the
“complements
on
tempered
coverings”
developed
in
[IUTchI],
§2
—
i.e.,
in
a
word,
as
a
consequence
of
the
“highly
anabelian
nature”
of
the
[full!]
tempered
fundamental
group
Π.
Theorem
2.2.
(Kummer-compatible
Multiradiality
of
Theta
Monoids)
Fix
a
collection
of
initial
Θ-data
(F
/F,
X
F
,
l,
C
K
,
V,
V
bad
mod
,
)
±ell
as
in
[IUTchI],
Definition
3.1.
Let
†
HT
Θ
NF
be
a
Θ
±ell
NF-Hodge
theater
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
Definition
6.13,
(i)].
For
∈
{,
×μ,
×μ},
write
Aut
F
(−)
for
the
group
of
automorphisms
of
the
F
-
prime-strip
in
parentheses
[cf.
[IUTchI],
Definition
5.2,
(iv);
[IUTchII],
Definition
4.9,
(vi),
(vii),
(viii)].
(i)
(Automorphisms
of
Prime-strips)
The
natural
functors
determined
by
assigning
to
an
F
-prime-strip
the
associated
F
×μ
-
and
F
×μ
-prime-strips
[cf.
[IUTchII],
Definition
4.9,
(vi),
(vii),
(viii)]
and
then
composing
with
the
natural
isomorphisms
of
Proposition
2.1,
(vi),
determine
natural
homomorphisms
†
×μ
†
(
D
>
))
Aut
F
×μ
(F
×μ
(
†
D
))
Aut
F
(F
env
(
D
>
))
→
Aut
F
×μ
(F
env
†
×μ
)
Aut
F
×μ
(
†
F
×μ
)
Aut
F
(
†
F
env
)
→
Aut
F
×μ
(
F
env
—
where
the
second
arrows
in
each
line
are
surjections
—
that
are
compatible
with
the
Kummer
isomorphisms
of
Proposition
2.1,
(ii),
and
Theorem
1.5,
(iii)
[cf.
the
final
portions
of
Proposition
2.1,
(iv),
(v),
(vi)].
(ii)
(Kummer
Aspects
of
Multiradiality
at
Bad
Primes)
Let
v
∈
V
bad
.
Write
⊥
†
∞
Ψ
env
(
D
>
)
v
⊆
∞
Ψ
env
(
†
D
>
)
v
;
Θ
⊥
†
∞
Ψ
F
env
(
HT
)
v
⊆
∞
Ψ
F
env
(
†
HT
Θ
)
v
for
the
submonoids
corresponding
to
the
respective
splittings
[cf.
[IUTchII],
Corol-
laries
3.5,
(iii);
3.6,
(iii)],
i.e.,
the
submonoids
generated
by
“
∞
θ
ι
env
(M
Θ
∗
)”
[cf.
the
notation
of
[IUTchII],
Proposition
3.1,
(i)]
and
the
respective
torsion
subgroups.
Now
consider
the
commutative
diagram
⊥
†
∞
Ψ
env
(
D
>
)
v
⊇
†
μ
∞
Ψ
env
(
D
>
)
v
⊆
†
×
∞
Ψ
env
(
D
>
)
v
Θ
⊥
†
∞
Ψ
F
env
(
HT
)
v
⊇
Θ
μ
†
∞
Ψ
F
env
(
HT
)
v
⊆
Θ
×
†
∞
Ψ
F
env
(
HT
)
v
⏐
⏐
⏐
⏐
⏐
⏐
†
×μ
∞
Ψ
env
(
D
>
)
v
→
∼
†
×μ
Ψ
ss
cns
(
D
)
v
⏐
⏐
Θ
×μ
†
∞
Ψ
F
env
(
HT
)
v
→
∼
†
×μ
Ψ
ss
cns
(
F
)
v
⏐
⏐
66
SHINICHI
MOCHIZUKI
—
where
the
inclusions
“⊇”,
“⊆”
are
the
natural
inclusions;
the
surjections
“”
are
the
natural
surjections;
the
superscript
“μ”
denotes
the
torsion
subgroup;
the
superscript
“×”
denotes
the
group
of
units;
the
superscript
“×μ”
denotes
the
quo-
tient
“(−)
×
/(−)
μ
”;
the
first
four
vertical
arrows
are
the
isomorphisms
determined
by
the
inverse
of
the
second
Kummer
isomorphism
of
the
third
display
of
Propo-
sition
2.1,
(ii);
†
D
is
as
discussed
in
Theorem
1.5,
(iii);
†
F
is
as
discussed
in
[IUTchII],
Corollary
4.10,
(i);
the
final
vertical
arrow
is
the
inverse
of
the
“Kummer
poly-isomorphism”
determined
by
the
second
displayed
isomorphism
of
[IUTchII],
Corollary
4.6,
(ii);
the
final
upper
horizontal
arrow
is
the
poly-
isomorphism
determined
by
composing
the
isomorphism
determined
by
the
in-
verse
of
the
second
displayed
natural
isomorphism
of
Proposition
2.1,
(vi),
with
the
†
×μ
induced
by
the
full
poly-automorphism
of
poly-automorphism
of
Ψ
ss
cns
(
D
)
v
†
the
D
-prime-strip
D
;
the
final
lower
horizontal
arrow
is
the
poly-automorphism
determined
by
the
condition
that
the
final
square
be
commutative.
This
commuta-
tive
diagram
is
compatible
with
the
various
group
actions
involved
relative
to
the
following
diagram
†
Π
X
(M
Θ
∗
(
D
>,v
))
†
G
v
(M
Θ
∗
(
D
>,v
))
=
†
G
v
(M
Θ
∗
(
D
>,v
))
=
†
G
v
(M
Θ
∗
(
D
>,v
))
→
∼
†
G
v
(M
Θ
∗
(
D
>,v
))
[cf.
the
notation
of
[IUTchII],
Proposition
3.1;
[IUTchII],
Remark
4.2.1,
(iv);
∼
[IUTchII],
Corollary
4.5,
(iv)]
—
where
“”
denotes
the
natural
surjection;
“
→
”
†
denotes
the
full
poly-automorphism
of
G
v
(M
Θ
∗
(
D
>,v
)).
Finally,
each
of
the
various
composite
maps
†
μ
ss
†
×μ
∞
Ψ
env
(
D
>
)
v
→
Ψ
cns
(
F
)
v
is
equal
to
the
zero
map
[cf.
(b
v
)
below;
the
final
portion
of
Proposition
2.1,
(iii)].
In
particular,
the
identity
automorphism
on
the
following
objects
is
compati-
ble,
relative
to
the
various
natural
morphisms
involved
[cf.
the
above
commutative
†
×μ
induced
by
arbi-
diagram],
with
the
collection
of
automorphisms
of
Ψ
ss
cns
(
F
)
v
trary
automorphisms
∈
Aut
F
×μ
(
†
F
×μ
)
[cf.
[IUTchII],
Corollary
1.12,
(iii);
[IUTchII],
Proposition
3.4,
(i)]:
(a
v
)
⊥
†
∞
Ψ
env
(
D
>
)
v
⊇
∞
Ψ
env
(
†
D
>
)
μ
v
;
†
(b
v
)
Π
μ
(M
Θ
∗
(
D
>,v
))
⊗
Q/Z
[cf.
the
discussion
of
Proposition
2.1,
(iii)],
rela-
∼
†
†
μ
tive
to
the
natural
isomorphism
Π
μ
(M
Θ
∗
(
D
>,v
))⊗Q/Z
→
∞
Ψ
env
(
D
>
)
v
of
[IUTchII],
Remark
1.5.2
[cf.
(a
v
)];
†
(c
v
)
the
projective
system
of
mono-theta
environments
M
Θ
∗
(
D
>,v
)
[cf.
(b
v
)];
†
†
μ
(d
v
)
the
splittings
∞
Ψ
⊥
env
(
D
>
)
v
∞
Ψ
env
(
D
>
)
v
[cf.
(a
v
)]
by
means
of
re-
striction
to
zero-labeled
evaluation
points
[cf.
[IUTchII],
Proposition
3.1,
(i)].
Proof.
The
various
assertions
of
Theorem
2.2
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
67
Remark
2.2.1.
In
light
of
the
central
importance
of
Theorem
2.2,
(ii),
in
the
theory
of
the
present
§2,
we
pause
to
examine
the
significance
of
Theorem
2.2,
(ii),
in
more
conceptual
terms.
def
(i)
In
the
situation
of
Theorem
2.2,
(ii),
let
us
write
[for
simplicity]
Π
v
=
def
def
†
Θ
†
Θ
†
Π
X
(M
Θ
∗
(
D
>,v
)),
G
v
=
G
v
(M
∗
(
D
>,v
)),
Π
μ
=
Π
μ
(M
∗
(
D
>,v
))
[cf.
(b
v
)].
Also,
def
†
for
simplicity,
we
write
(l
·
Δ
Θ
)
=
(l
·
Δ
Θ
)(M
Θ
∗
(
D
>,v
))
[cf.
[IUTchII],
Proposition
1.5,
(iii)].
Here,
we
recall
that
in
fact,
(l
·
Δ
Θ
)
may
be
thought
of
as
an
object
constructed
from
Π
v
[cf.
[IUTchII],
Proposition
1.4].
Then
the
projective
system
†
of
mono-theta
environments
M
Θ
∗
(
D
>,v
)
[cf.
(c
v
)]
may
be
thought
of
as
a
sort
of
“amalgamation
of
Π
v
and
Π
μ
”,
where
the
amalgamation
is
such
that
it
allows
the
reconstruction
of
the
mono-theta-theoretic
cyclotomic
rigidity
isomorphism
∼
(l
·
Δ
Θ
)
→
Π
μ
×
-orbit
of
this
isomor-
[cf.
[IUTchII],
Proposition
1.5,
(iii)]
—
i.e.,
not
just
the
Z
phism!
†
(ii)
Now,
in
the
notation
of
(i),
the
Kummer
classes
∈
∞
Ψ
⊥
env
(
D
>
)
v
[cf.
(a
v
)]
constituted
by
the
various
étale
theta
functions
may
be
thought
of,
for
a
suitable
characteristic
open
subgroup
H
⊆
Π
v
,
as
twisted
homomorphisms
(Π
v
⊇)
H
→
Π
μ
whose
restriction
to
(l
·
Δ
Θ
)
coincides
with
the
cyclotomic
rigidity
isomorphism
∼
(l
·
Δ
Θ
)
→
Π
μ
discussed
in
(i).
Then
the
essential
content
of
Theorem
2.2,
(ii),
lies
in
the
observation
that
since
the
Kummer-theoretic
link
between
étale-like
data
and
Frobenius-
like
data
at
v
∈
V
bad
is
established
by
means
of
projective
systems
of
mono-theta
environments
[cf.
the
discussion
of
Proposition
2.1,
(iii)]
—
i.e.,
which
do
not
involve
the
various
monoids
“(−)
×μ
”!
—
the
mono-
theta-theoretic
cyclotomic
rigidity
isomorphism
[i.e.,
not
just
the
×
-orbit
of
this
isomorphism!]
is
immune
to
the
various
automorphisms
Z
of
the
monoids
“(−)
×μ
”
which,
from
the
point
of
view
of
the
multiradial
formulation
to
be
discussed
in
Corollary
2.3
below,
arise
from
isomor-
phisms
of
coric
data.
Put
another
way,
this
“immunity”
may
be
thought
of
as
a
sort
of
decoupling
of
the
“geometric”
[i.e.,
in
the
sense
of
the
geometric
fundamental
group
Δ
v
⊆
Π
v
]
and
“base-field-theoretic”
[i.e.,
associated
to
the
local
absolute
Galois
group
Π
v
G
v
]
data
which
allows
one
to
treat
the
exterior
cyclotome
Π
μ
—
which,
a
priori,
“looks
base-field-theoretic”
—
as
being
part
of
the
“geometric”
data.
From
the
point
of
view
of
the
multiradial
formulation
to
be
discussed
in
Corollary
2.3
below
[cf.
also
the
discussion
of
[IUTchII],
Remark
1.12.2,
(vi)],
this
decoupling
may
be
thought
of
as
a
sort
of
splitting
into
purely
radial
and
purely
coric
components
—
i.e.,
with
respect
to
which
Π
μ
is
“purely
radial”,
while
the
various
monoids
“(−)
×μ
”
are
“purely
coric”.
68
SHINICHI
MOCHIZUKI
(iii)
Note
that
the
immunity
to
automorphisms
of
the
monoids
“(−)
×μ
”
dis-
×
-indeterminacies
that
arise
in
the
case
cussed
in
(ii)
lies
in
stark
contrast
to
the
Z
of
the
cyclotomic
rigidity
isomorphisms
constructed
from
MLF-Galois
pairs
in
a
fashion
that
makes
essential
use
of
the
monoids
“(−)
×μ
”,
as
discussed
in
[IUTchII],
Corollary
1.11;
[IUTchII],
Remark
1.11.3.
In
the
following
discussion,
let
us
write
“O
×μ
”
for
the
various
monoids
“(−)
×μ
”
that
occur
in
the
situation
of
Theorem
”
[cf.
the
2.2;
also,
we
shall
use
similar
notation
“O
μ
”,
“O
×
”,
“O
”,
“O
gp
”,
“O
gp
notational
conventions
of
[IUTchII],
Example
1.8,
(ii),
(iii),
(iv),
(vii)].
Thus,
we
have
a
diagram
O
μ
⊆
O
×
⏐
⏐
⊆
O
⊆
O
gp
⊆
O
gp
O
×μ
of
natural
morphisms
between
monoids
equipped
with
Π
v
-actions.
Relative
to
this
notation,
the
essential
input
data
for
the
cyclotomic
rigidity
isomorphism
con-
structed
from
an
MLF-Galois
pair
is
given
by
“O
”
[cf.
[IUTchII],
Corollary
1.11,
×
-indeterminacy
act-
(a)].
On
the
other
hand
—
unlike
the
case
with
O
μ
—
a
Z
×
-
ing
on
O
×μ
does
not
lie
under
an
identity
action
on
O
×
!
That
is
to
say,
a
Z
×
-indeterminacies
on
indeterminacy
acting
on
O
×μ
can
only
be
lifted
naturally
to
Z
[cf.
Fig.
2.1
below;
[IUTchII],
Corollary
1.11,
(a),
in
the
case
where
one
O
×
,
O
gp
×
-
×
;
[IUTchII],
Remark
1.11.3,
(ii)].
In
the
presence
of
such
Z
takes
“Γ”
to
be
Z
×
-orbit
of
the
MLF-Galois-pair-theoretic
indeterminacies,
one
can
only
recover
the
Z
cyclotomic
rigidity
isomorphism.
×
Z
×
Z
×
Z
O
×μ
O
×
⊆
O
⊆
O
gp
⊆
O
gp
(⊇
O
μ
)
×
-indeterminacies
in
the
case
of
Fig.
2.1:
Induced
Z
MLF-Galois
pair
cyclotomic
rigidity
×
Z
id
∼
Π
μ
→
O
μ
→
O
×μ
×
-indeterminacies
in
the
case
of
Fig.
2.2:
Insulation
from
Z
mono-theta-theoretic
cyclotomic
rigidity
(iv)
Thus,
in
summary,
[cf.
Fig.
2.2
above]
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
69
mono-theta-theoretic
cyclotomic
rigidity
plays
an
essential
role
in
the
theory
of
the
present
§2
—
and,
indeed,
in
the
theory
of
the
present
series
of
papers!
—
in
that
it
serves
to
insulate
the
étale
theta
function
×
-indeterminacies
which
act
on
the
coric
log-shells
[i.e.,
the
from
the
Z
various
monoids
“(−)
×μ
”].
The
techniques
that
underlie
the
resulting
multiradiality
of
theta
monoids
[cf.
Corol-
lary
2.3
below],
cannot,
however,
be
applied
immediately
to
the
case
of
Gaussian
monoids.
That
is
to
say,
the
corresponding
multiradiality
of
Gaussian
monoids,
to
be
discussed
in
§3
below,
requires
one
to
apply
the
theory
of
log-shells
developed
in
§1
[cf.
[IUTchII],
Remark
2.9.1,
(iii);
[IUTchII],
Remark
3.4.1,
(ii);
[IUTchII],
Remark
3.7.1].
On
the
other
hand,
as
we
shall
see
in
§3
below,
the
multiradiality
of
Gaussian
monoids
depends
in
an
essential
way
on
the
multiradiality
of
theta
monoids
discussed
in
the
present
§2
as
a
sort
of
“essential
first
step”
constituted
by
the
decoupling
discussed
in
(ii)
above.
Indeed,
if
one
tries
to
consider
the
Kummer
j
2
theory
of
the
theta
values
[i.e.,
the
“q
”
—
cf.
[IUTchII],
Remark
2.5.1,
(i)]
just
v
as
elements
of
the
base
field
—
i.e.,
without
availing
oneself
of
the
theory
of
the
étale
theta
function
—
then
it
is
difficult
to
see
how
to
rigidify
the
cyclotomes
involved
by
any
means
other
than
the
theory
of
MLF-Galois
pairs
discussed
in
(iii)
above.
But,
as
discussed
in
(iii)
above,
this
approach
to
cyclotomic
rigidity
gives
rise
to
2
×
-indeterminacies
—
i.e.,
to
confusion
between
the
theta
values
“q
j
”
and
their
Z
v
×
-powers,
which
is
unacceptable
from
the
point
of
view
of
the
theory
of
the
present
Z
series
of
papers!
For
another
approach
to
understanding
the
indispensability
of
the
multiradiality
of
theta
monoids,
we
refer
to
Remark
2.2.2
below.
Remark
2.2.2.
(i)
One
way
to
understand
the
very
special
role
played
by
the
theta
values
[i.e.,
the
values
of
the
theta
function]
in
the
theory
of
the
present
series
of
papers
is
to
consider
the
following
naive
question:
Can
one
develop
a
similar
theory
to
the
theory
of
the
present
series
of
papers
in
which
one
replaces
the
Θ
×μ
gau
-link
q
→
1
2
..
.
2
)
(l
q
[cf.
[IUTchII],
Remark
4.11.1]
by
a
correspondence
of
the
form
q
→
q
λ
—
where
λ
is
some
arbitrary
positive
integer?
The
answer
to
this
question
is
“no”.
Indeed,
such
a
correspondence
does
not
come
equipped
with
the
extensive
multiradiality
machinery
—
such
as
mono-theta-
theoretic
cyclotomic
rigidity
and
the
splittings
determined
by
zero-labeled
70
SHINICHI
MOCHIZUKI
evaluation
points
—
that
has
been
developed
for
the
étale
theta
function
[cf.
the
discussion
of
Step
(vi)
of
the
proof
of
Corollary
3.12
of
§3
below].
For
instance,
the
lack
of
mono-theta-theoretic
cyclotomic
rigidity
means
that
one
does
not
have
an
apparatus
for
insulating
the
Kummer
classes
of
such
a
correspondence
from
the
×
-indeterminacies
that
act
on
the
various
monoids
“(−)
×μ
”
[cf.
the
discussion
Z
of
Remark
2.2.1,
(iv)].
The
splittings
determined
by
zero-labeled
evaluation
points
also
play
an
essential
role
in
decoupling
these
monoids
“(−)
×μ
”
—
i.e.,
the
coric
log-shells
—
from
the
“purely
radial”
[or,
put
another
way,
“value
group”]
portion
of
such
a
correspondence
“q
→
q
λ
”
[cf.
the
discussion
of
(iii)
below;
Remark
2.2.1,
(ii);
[IUTchII],
Remark
1.12.2,
(vi)].
Note,
moreover,
that
if
one
tries
to
realize
such
a
multiradial
splitting
via
evaluation
—
i.e.,
in
accordance
with
the
principle
of
“Galois
evaluation”
[cf.
the
discussion
of
[IUTchII],
Remark
1.12.4]
—
for
a
correspondence
“q
→
q
λ
”
by,
for
instance,
taking
λ
to
be
one
of
the
“j
2
”
[where
j
is
a
positive
integer]
that
appears
as
a
value
of
the
étale
theta
function,
then
one
must
contend
with
issues
of
symmetry
between
the
zero-labeled
evaluation
point
and
the
evaluation
point
corresponding
to
λ
—
i.e.,
symmetry
issues
that
are
resolved
in
the
theory
of
the
present
series
of
papers
by
means
of
the
theory
surrounding
the
F
±
l
-symmetry
[cf.
the
discussion
of
[IUTchII],
Remarks
2.6.2,
3.5.2].
As
discussed
in
[IUTchII],
Remark
2.6.3,
this
sort
of
situation
leads
to
numerous
conditions
on
the
collection
of
evaluation
points
under
consideration.
In
particular,
ultimately,
it
is
difficult
to
see
how
to
construct
a
theory
as
in
the
present
series
of
papers
for
any
collection
of
evaluation
points
other
than
the
collection
that
is
in
fact
adopted
in
the
definition
of
the
Θ
×μ
gau
-link.
(ii)
As
discussed
in
Remark
2.2.1,
(iv),
we
shall
be
concerned,
in
§3
below,
with
developing
multiradial
formulations
for
Gaussian
monoids.
These
multiradial
for-
mulations
will
be
subject
to
certain
indeterminacies,
which
—
although
sufficiently
mild
to
allow
the
execution
of
the
volume
computations
that
will
be
the
subject
of
[IUTchIV]
—
are,
nevertheless,
substantially
more
severe
than
the
indeterminacies
that
occur
in
the
multiradial
formulation
given
for
theta
monoids
in
the
present
§2
[cf.
Corollary
2.3
below].
Indeed,
the
indeterminacies
in
the
multiradial
formulation
given
for
theta
monoids
in
the
present
§2
—
which
essentially
consist
of
multiplica-
tion
by
roots
of
unity
[cf.
[IUTchII],
Proposition
3.1,
(i)]
—
are
essentially
negligible
and
may
be
regarded
as
a
consequence
of
the
highly
nontrivial
Kummer
theory
surrounding
mono-theta
environments
[cf.
Proposition
2.1,
(iii);
Theorem
2.2,
(ii)],
which,
as
discussed
in
Remark
2.2.1,
(iv),
cannot
be
mimicked
for
“theta
val-
ues
regarded
just
as
elements
of
the
base
field”.
That
is
to
say,
the
quite
exact
nature
of
the
multiradial
formulation
for
theta
monoids
—
i.e.,
which
contrasts
sharply
with
the
somewhat
approximate
nature
of
the
multiradial
formulation
for
Gaussian
monoids
to
be
developed
in
§3
—
constitutes
another
important
ingre-
dient
of
the
theory
of
the
present
paper
that
one
must
sacrifice
if
one
attempts
to
work
with
correspondences
q
→
q
λ
as
discussed
in
(i),
i.e.,
correspondences
which
do
not
come
equipped
with
the
extensive
multiradiality
machinery
that
arises
as
a
consequence
of
the
theory
of
the
étale
theta
function
developed
in
[EtTh].
(iii)
One
way
to
understand
the
significance,
in
the
context
of
the
discussions
of
(i)
and
(ii)
above,
of
the
multiradial
coric/radial
decouplings
furnished
by
the
splittings
determined
by
the
zero-labeled
evaluation
points
is
as
follows.
Ultimately,
in
order
to
establish,
in
§3
below,
multiradial
formulations
for
Gaussian
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
71
monoids,
it
will
be
of
crucial
importance
to
pass
from
the
Frobenius-like
theta
monoids
that
appear
in
the
domain
of
the
Θ
×μ
gau
-link
to
vertically
coric
étale-
like
objects
by
means
of
Kummer
theory
[cf.
the
discussions
of
Remarks
1.2.4,
(i);
1.5.4,
(i),
(iii)],
in
the
context
of
the
relevant
log-Kummer
correspondences,
as
discussed,
for
instance,
in
Remark
3.12.2,
(iv),
(v),
below
[cf.
also
[IUTchII],
Remark
1.12.2,
(iv)].
On
the
other
hand,
in
order
to
obtain
formulations
expressed
in
terms
that
are
meaningful
from
the
point
of
view
of
the
codomain
of
the
Θ
×μ
gau
-
link,
it
is
necessary
[cf.
the
discussion
of
Remark
3.12.2,
(iv),
(v),
below]
to
relate
this
Kummer
theory
of
theta
monoids
in
the
domain
of
the
Θ
×μ
gau
-link
to
the
Kummer
theory
constituted
by
the
×μ-Kummer
structures
that
appear
in
the
horizontally
coric
portion
of
the
data
that
constitutes
the
Θ
×μ
gau
-link
[cf.
Theorem
1.5,
(ii)].
This
is
precisely
what
is
achieved
by
the
Kummer-compatibility
of
the
multiradial
splitting
via
evaluation
—
i.e.,
in
accordance
with
the
principle
of
“Galois
evaluation”
[cf.
the
discussion
of
[IUTchII],
Remark
1.12.4].
This
state
of
affairs
[cf.,
especially,
the
two
displays
of
[IUTchII],
Corollary
1.12,
(ii);
the
final
arrow
of
the
diagram
“(†
μ,×μ
)”
of
[IUTchII],
Corollary
1.12,
(iii)]
is
illustrated
in
Fig.
2.3
below.
id
∞
θ
Aut(G),
Ism
Π
←
Π/Δ
→
G
O
×μ
id
Aut(G),
Ism
O
×
·
∞
θ
Π
←
→
Π/Δ
..
.
G
O
×μ
→
∞
θ
→
1
∈
O
×μ
Fig.
2.3:
Kummer-compatible
splittings
via
evaluation
at
zero-labeled
evaluation
points
[i.e.,
“Π
←
Π/Δ”]
Here,
the
multiple
arrows
[i.e.,
indicated
by
means
of
the
“→’s”
separated
by
vertical
dots]
in
the
lower
portion
of
the
diagram
correspond
to
the
fact
that
the
“O
×
”
on
the
left-hand
side
of
this
lower
portion
is
related
to
the
“O
×μ
”
on
the
right-
hand
side
via
an
Ism-orbit
of
morphisms;
the
analogous
arrow
in
the
upper
portion
of
the
diagram
consists
of
a
single
arrow
[i.e.,
“→”]
and
corresponds
to
the
fact
that
the
restriction
of
the
multiple
arrows
in
the
lower
portion
of
the
diagram
to
“
∞
θ”
amounts
to
a
single
arrow,
i.e.,
precisely
as
a
consequence
of
the
fact
that
×μ
[cf.
the
situation
illustrated
in
Fig.
2.2].
On
the
other
hand,
∞
θ
→
1
∈
O
the
“Π/Δ’s”
on
the
left-hand
side
of
both
the
upper
and
the
lower
portions
of
the
72
SHINICHI
MOCHIZUKI
diagram
are
related
to
the
“G’s”
on
the
right-hand
side
via
the
unique
tautological
Aut(G)-orbit
of
isomorphisms.
Thus,
from
the
point
of
view
of
Fig.
2.3,
the
crucial
Kummer-compatibility
discussed
above
may
be
understood
as
the
statement
that
the
multiradial
structure
[cf.
the
lower
portion
of
Fig.
2.3]
on
the
“theta
monoid
O
×
·
∞
θ”
furnished
by
the
splittings
via
Galois
evaluation
into
coric/radial
components
is
compatible
with
the
relationship
between
the
respective
Kummer
theories
of
the
“O
×
”
portion
of
“O
×
·
∞
θ”
[on
the
left]
and
the
coric
“O
×μ
”
[on
the
right].
This
state
of
affairs
lies
in
stark
contrast
to
the
situation
that
arises
in
the
case
of
a
naive
correspondence
of
the
form
“q
→
q
λ
”
as
discussed
in
(i):
That
is
to
say,
in
the
case
of
such
a
naive
correspondence,
the
corresponding
arrows
“→”
of
the
analogue
of
Fig.
2.3
map
q
λ
→
1
∈
O
×μ
and
hence
are
fundamentally
incompatible
with
passage
to
Kummer
classes,
i.e.,
since
the
Kummer
class
of
q
λ
in
a
suitable
cohomology
group
of
Π/Δ
is
by
no
∼
means
mapped,
via
the
poly-isomorphism
Π/Δ
→
G,
to
the
trivial
element
of
the
relevant
cohomology
group
of
G.
We
conclude
the
present
§2
with
the
following
multiradial
interpretation
[cf.
[IUTchII],
Remark
4.1.1,
(iii);
[IUTchII],
Remark
4.3.1]
—
in
the
spirit
of
the
étale-
picture
of
D-Θ
±ell
NF-Hodge
theaters
of
[IUTchII],
Corollary
4.11,
(ii)
—
of
the
theory
surrounding
Theorem
2.2.
Corollary
2.3.
(Étale-picture
of
Multiradial
Theta
Monoids)
In
the
notation
of
Theorem
2.2,
let
{
n,m
HT
Θ
±ell
NF
}
n,m∈Z
be
a
collection
of
distinct
Θ
±ell
NF-Hodge
theaters
[relative
to
the
given
initial
Θ-data]
—
which
we
think
of
as
arising
from
a
Gaussian
log-theta-lattice
[cf.
±ell
Definition
1.4].
Write
n,m
HT
D-Θ
NF
for
the
D-Θ
±ell
NF-Hodge
theater
associated
±ell
to
n,m
HT
Θ
NF
.
Consider
the
radial
environment
[cf.
[IUTchII],
Example
1.7,
(ii)]
defined
as
follows.
We
define
a
collection
of
radial
data
†
±ell
R
=
(
†
HT
D-Θ
NF
∼
×μ
†
†
†
bad
†
,
F
,
F
×μ
(
†
D
),
F
×μ
(
D
))
env
(
D
>
),
R
env
(
D
>
)
→
F
to
consist
of
±ell
(a
R
)
a
D-Θ
±ell
NF-Hodge
theater
†
HT
D-Θ
NF
;
±ell
D-Θ
†
†
(b
R
)
the
F
-prime-strip
F
env
(
D
>
)
associated
to
HT
tion
2.1,
(ii)];
NF
[cf.
Proposi-
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
73
(c
R
)
the
data
(a
v
),
(b
v
),
(c
v
),
(d
v
)
of
Theorem
2.2,
(ii),
for
v
∈
V
bad
,
which
we
denote
by
†
R
bad
;
±ell
(d
R
)
the
F
×μ
-prime-strip
F
×μ
(
†
D
)
associated
to
†
HT
D-Θ
orem
1.5,
(iii)];
NF
[cf.
The-
∼
×μ
†
†
(e
R
)
the
full
poly-isomorphism
of
F
×μ
-prime-strips
F
×μ
(
D
).
env
(
D
>
)
→
F
We
define
a
morphism
between
two
collections
of
radial
data
†
R
→
‡
R
[where
we
apply
the
evident
notational
conventions
with
respect
to
“†”
and
“‡”]
to
consist
of
data
as
follows:
±ell
(a
Mor
R
)
an
isomorphism
of
D-Θ
±ell
NF-Hodge
theaters
†
HT
D-Θ
NF
∼
‡
→
HT
D-Θ
±ell
NF
∼
†
‡
(b
Mor
R
)
the
isomorphism
of
F
-prime-strips
F
env
(
D
>
)
→
F
env
(
D
>
)
induced
by
the
isomorphism
of
(a
Mor
R
);
∼
(c
Mor
R
)
the
isomorphism
between
collections
of
data
†
R
bad
→
‡
R
bad
induced
by
the
isomorphism
of
(a
Mor
R
);
∼
(d
Mor
R
)
an
isomorphism
of
F
×μ
-prime-strips
F
×μ
(
†
D
)
→
F
×μ
(
‡
D
);
(e
Mor
R
)
we
observe
that
the
isomorphisms
of
(b
Mor
R
)
and
(d
Mor
R
)
are
necessarily
compatible
with
the
poly-isomorphisms
of
(e
R
)
for
“†”,
“‡”.
We
define
a
collection
of
coric
data
†
C
=
(
†
D
,
F
×μ
(
†
D
))
to
consist
of
(a
C
)
a
D
-prime-strip
†
D
;
(b
C
)
the
F
×μ
-prime-strip
F
×μ
(
†
D
)
associated
to
†
D
[cf.
[IUTchII],
Corollary
4.5,
(ii);
[IUTchII],
Definition
4.9,
(vi),
(vii)].
We
define
a
morphism
between
two
collections
of
coric
data
†
C
→
‡
C
[where
we
apply
the
evident
notational
conventions
with
respect
to
“†”
and
“‡”]
to
consist
of
data
as
follows:
∼
(a
Mor
C
)
an
isomorphism
of
D
-prime-strips
†
D
→
‡
D
;
∼
(b
Mor
C
)
an
isomorphism
of
F
×μ
-prime-strips
F
×μ
(
†
D
)
→
F
×μ
(
‡
D
)
that
∼
induces
the
isomorphism
†
D
→
‡
D
on
associated
D
-prime-strips
of
(a
Mor
C
).
The
radial
algorithm
is
given
by
the
assignment
†
±ell
R
=
(
†
HT
D-Θ
NF
∼
×μ
†
†
†
bad
†
,F
,
F
×μ
(
†
D
),
F
×μ
(
D
))
env
(
D
>
),
R
env
(
D
>
)
→
F
→
†
C
=
(
†
D
,
F
×μ
(
†
D
))
;
74
SHINICHI
MOCHIZUKI
—
together
with
the
assignment
on
morphisms
determined
by
the
data
of
(d
Mor
R
).
Then:
(i)
The
functor
associated
to
the
radial
algorithm
defined
above
is
full
and
essentially
surjective.
In
particular,
the
radial
environment
defined
above
is
multiradial.
±ell
(ii)
Each
D-Θ
±ell
NF-Hodge
theater
n,m
HT
D-Θ
NF
,
for
n,
m
∈
Z,
defines,
in
an
evident
way,
an
associated
collection
of
radial
data
n,m
R.
The
poly-isomorphisms
induced
by
the
vertical
arrows
of
the
Gaussian
log-theta-lattice
under
consid-
∼
eration
[cf.
Theorem
1.5,
(i)]
induce
poly-isomorphisms
of
radial
data
.
.
.
→
n,m
R
∼
∼
→
n,m+1
R
→
.
.
.
.
Write
n,◦
R
for
the
collection
of
radial
data
obtained
by
identifying
the
various
n,m
R,
for
m
∈
Z,
via
these
poly-isomorphisms
and
n,◦
C
for
the
collection
of
coric
data
associated,
via
the
radial
algorithm
defined
above,
to
the
radial
data
n,◦
R.
In
a
similar
vein,
the
horizontal
arrows
of
the
Gaussian
log-theta-lattice
under
consideration
induce
∼
∼
∼
full
poly-isomorphisms
.
.
.
→
n,m
D
→
n+1,m
D
→
.
.
.
of
D
-prime-strips
[cf.
Theorem
1.5,
(ii)].
Write
◦,◦
C
for
the
collection
of
coric
data
obtained
by
identifying
the
various
n,◦
C,
for
n
∈
Z,
via
these
poly-isomorphisms.
Thus,
by
applying
the
radial
algorithm
defined
above
to
each
n,◦
R,
for
n
∈
Z,
we
obtain
a
diagram
—
i.e.,
an
étale-picture
of
radial
data
—
as
in
Fig.
2.4
below.
This
diagram
satisfies
the
important
property
of
admitting
arbitrary
permutation
symmetries
among
the
spokes
[i.e.,
the
labels
n
∈
Z]
and
is
compatible,
in
the
evident
sense,
with
the
étale-picture
of
D-Θ
±ell
NF-Hodge
theaters
of
[IUTchII],
Corollary
4.11,
(ii).
n,◦
F
D
>
)
env
(
n,◦
bad
+
R
+
...
...
n
,◦
F
D
>
)
env
(
n
,◦
bad
+
R
+
...
...
...
|
—
×μ
◦,◦
F
(
D
)
—
|
n
,◦
F
D
>
)
env
(
n
,◦
bad
+
R
+
...
Fig.
2.4:
Étale-picture
of
radial
data
n
,◦
F
D
>
)
env
(
n
,◦
bad
+
R
+
...
...
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
75
(iii)
The
[poly-]isomorphisms
of
F
×μ
-prime-strips
of/induced
by
(e
R
),
(b
Mor
R
),
(d
Mor
R
)
[cf.
also
(e
Mor
R
)]
are
compatible,
relative
to
the
Kummer
isomor-
phisms
of
Proposition
2.1,
(ii)
[cf.
also
Proposition
2.1,
(vi)],
and
Theorem
1.5,
(iii),
with
the
poly-isomorphisms
—
arising
from
the
horizontal
arrows
of
the
Gaussian
log-theta-lattice
—
of
Theorem
1.5,
(ii).
∼
†
‡
(iv)
The
algorithmic
construction
of
the
isomorphisms
F
env
(
D
>
)
→
F
env
(
D
>
),
∼
†
bad
R
→
‡
R
bad
of
(b
Mor
R
),
(c
Mor
R
),
as
well
as
of
the
Kummer
isomorphisms
and
poly-isomorphisms
of
projective
systems
of
mono-theta
environments
discussed
in
Proposition
2.1,
(ii),
(iii)
[cf.
also
Proposition
2.1,
(vi);
the
second
display
of
Theorem
2.2,
(ii)],
and
Theorem
1.5,
(iii),
(v),
are
compatible
[cf.
the
final
portions
of
Theorems
1.5,
(v);
2.2,
(ii)]
with
the
horizontal
arrows
of
the
Gaussian
log-theta-lattice
[cf.,
e.g.,
the
full
poly-isomorphisms
of
Theorem
1.5,
(ii)],
in
the
sense
that
these
constructions
are
stabilized/equivariant/functorial
with
respect
to
arbitrary
automomorphisms
of
the
domain
and
codomain
of
these
horizontal
arrows
of
the
Gaussian
log-theta-lattice.
Proof.
The
various
assertions
of
Corollary
2.3
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
2.3.1.
(i)
In
the
context
of
the
étale-picture
of
Fig.
2.4,
it
is
of
interest
to
recall
the
point
of
view
of
the
discussion
of
[IUTchII],
1.12.5,
(i),
(ii),
concerning
the
analogy
between
étale-pictures
in
the
theory
of
the
present
series
of
papers
and
the
polar
coordinate
representation
of
the
classical
Gaussian
integral.
(ii)
The
étale-picture
discussed
in
Corollary
2.3,
(ii),
may
be
thought
of
as
a
sort
of
canonical
splitting
of
the
portion
of
the
Gaussian
log-theta-lattice
under
consideration
that
involves
theta
monoids
[cf.
the
discussion
of
[IUTchI],
§I1,
preceding
Theorem
A].
(iii)
The
portion
of
the
multiradiality
discussed
in
Corollary
2.3,
(iv),
at
v
∈
V
bad
corresponds,
in
essence,
to
the
multiradiality
discussed
in
[IUTchII],
Corollary
1.12,
(iii);
[IUTchII],
Proposition
3.4,
(i).
Remark
2.3.2.
A
similar
result
to
Corollary
2.3
may
be
formulated
concern-
ing
the
multiradiality
properties
satisfied
by
the
Kummer
theory
of
∞
κ-coric
structures
as
discussed
in
[IUTchII],
Corollary
4.8.
That
is
to
say,
the
Kummer
theory
of
the
localization
poly-morphisms
κ-sol
†
†
†
†
{π
1
(
D
)
M
∞
κ
}
j
→
M
∞
κv
j
⊆
M
∞
κ×v
j
v∈V
discussed
in
[IUTchII],
Corollary
4.8,
(iii),
is
based
on
the
cyclotomic
rigidity
isomorphisms
for
∞
κ-coric
structures
discussed
in
[IUTchI],
Example
5.1,
(v);
[IUTchI],
Definition
5.2,
(vi),
(viii)
[cf.
also
the
discussion
of
[IUTchII],
Corol-
lary
4.8,
(i)],
which
satisfy
“insulation”
properties
analogous
to
the
properties
discussed
in
Remark
2.2.1
in
the
case
of
mono-theta-theoretic
cyclotomic
rigidity.
76
SHINICHI
MOCHIZUKI
Moreover,
the
reconstruction
of
∞
κ-coric
structures
from
∞
κ×-structures
via
restriction
of
Kummer
classes
‡
∼
M
∞
κv
j
⊆
‡
M
∞
κ×v
j
→
‡
M
×
→
‡
M
×
v
j
∞
κ×v
j
as
discussed
in
[IUTchI],
Definition
5.2,
(vi),
(viii)
—
i.e.,
a
reconstruction
in
ac-
cordance
with
the
principle
of
Galois
evaluation
[cf.
[IUTchII],
Remark
1.12.4]
—
may
be
regarded
as
a
decoupling
into
}
;
†
M
∞
κv
j
;
‡
M
∞
κv
j
]
and
·
radial
[i.e.,
{π
1
κ-sol
(
†
D
)
†
M
∞
κ
j
∼
→
‡
M
×
·
coric
[i.e.,
the
quotient
of
‡
M
×
v
j
by
its
torsion
subgroup]
∞
κ×v
j
components,
i.e.,
in
an
entirely
analogous
fashion
to
the
mono-theta-theoretic
case
discussed
in
Remark
2.2.2,
(iii).
The
Galois
evaluation
that
gives
rise
to
the
j
2
theta
values
“q
”
in
the
case
of
theta
monoids
corresponds
to
the
construction
via
v
Galois
evaluation
of
the
monoids
“
†
M
mod
”,
i.e.,
via
the
operation
of
restricting
Kummer
classes
associated
to
elements
of
∞
κ-coric
structures,
as
discussed
in
[IUTchI],
Example
5.1,
(v);
[IUTchII],
Corollary
4.8,
(i)
[cf.
also
[IUTchI],
Defini-
tion
5.2,
(vi),
(viii)].
We
leave
the
routine
details
of
giving
a
formulation
in
the
style
of
Corollary
2.3
to
the
reader.
Remark
2.3.3.
In
the
context
of
Remark
2.3.2,
it
is
of
interest
to
compare
and
contrast
the
multiradiality
properties
that
hold
in
the
theta
[cf.
Remarks
2.2.1,
2.2.2;
Corollary
2.3]
and
number
field
[cf.
Remark
2.3.2]
cases,
as
follows.
(i)
One
important
similarity
between
the
theta
and
number
field
cases
lies
in
the
establishment
of
multiradiality
properties,
i.e.,
such
as
the
radial/coric
decou-
pling
discussed
in
Remarks
2.2.2,
(iii);
2.3.2,
by
using
the
geometric
dimension
of
the
elliptic
curve
under
consideration
as
a
sort
of
“multiradial
geometric
container”
for
the
radial
arithmetic
data
j
2
of
interest,
i.e.,
theta
values
“q
”
or
copies
of
the
number
field
“F
mod
”.
v
That
is
to
say,
in
the
theta
case,
the
theory
of
theta
functions
on
Tate
curves
as
developed
in
[EtTh]
furnishes
such
a
geometric
container
for
the
theta
values,
while
in
the
number
field
case,
the
absolute
anabelian
interpretation
developed
in
[AbsTopIII]
of
the
theory
of
Belyi
maps
as
Belyi
cuspidalizations
[cf.
[IUTchI],
Remark
5.1.4]
furnishes
such
a
geometric
container
for
copies
of
F
mod
.
In
this
context,
another
important
similarity
is
the
passage
from
such
a
geometric
container
to
the
radial
arithmetic
data
of
interest
by
means
of
Galois
evaluation
[cf.
Remark
2.2.2,
(i),
(iii);
Remark
2.3.2].
(ii)
One
important
theme
of
the
present
series
of
papers
is
the
point
of
view
of
dismantling
the
two
underlying
combinatorial
dimensions
of
[the
ring
of
inte-
gers
of]
a
number
field
—
cf.
the
discussion
of
Remark
3.12.2
below.
As
discussed
in
[IUTchI],
Remark
6.12.3
[cf.
also
[IUTchI],
Remark
6.12.6],
this
dismantling
may
be
compared
to
the
dismantling
of
the
single
complex
holomorphic
dimension
of
the
upper
half-plane
into
two
underlying
real
dimensions.
If
one
considers
this
dismantling
from
such
a
classical
point
of
view,
then
one
is
tempted
to
attempt
to
understand
the
dismantling
into
two
underlying
real
dimensions,
by,
in
effect,
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
77
base-changing
from
R
to
C,
so
as
to
obtain
two-dimensional
complex
holomorphic
objects,
which
we
regard
as
being
equipped
with
some
sort
of
descent
data
arising
from
the
base-change
from
R
to
C.
Translating
this
approach
back
into
the
case
of
number
fields,
one
obtains
a
situ-
ation
in
which
one
attempts
to
understand
the
dismantling
of
the
two
underlying
combinatorial
dimensions
of
[the
ring
of
integers
of]
a
number
field
by
working
with
two-dimensional
scheme-theoretic
data
—
i.e.,
such
as
an
elliptic
curve
over
[a
suitable
localization
of
the
ring
of
integers
of]
a
number
field
—
equipped
with
“suitable
descent
data”.
From
this
point
of
view,
one
may
think
of
the
“multiradial
geometric
containers”
discussed
in
(i)
as
a
sort
of
realization
of
such
two-dimensional
scheme-theoretic
data,
and
of
the
accompanying
Galois
evaluation
operations,
i.e.,
the
multiradial
representations
up
to
certain
mild
indeterminacies
obtained
in
The-
orem
3.11,
below
[cf.
also
the
discussion
of
Remark
3.12.2,
below],
as
a
sort
of
realization
of
the
corresponding
“suitable
descent
data”.
This
sort
of
interpretation
is
reminiscent
of
the
interpretation
of
multiradiality
in
terms
of
parallel
transport
via
a
connection
as
discussed
in
[IUTchII],
Remark
1.7.1,
and
the
closely
related
interpretation
given
in
the
discussion
of
[IUTchII],
Remark
1.9.2,
(iii),
of
the
tautological
approach
to
multiradiality
in
terms
of
PD-envelopes
in
the
style
of
the
p-adic
theory
of
the
crystalline
site.
(iii)
Another
fundamental
similarity
between
the
theta
and
number
field
cases
may
be
seen
in
the
fact
that
the
associated
Galois
evaluation
operations
—
i.e.,
j
2
that
give
rise
to
the
theta
values
“q
”
[cf.
[IUTchII],
Corollary
3.6]
or
copies
of
the
v
number
field
“F
mod
”
[cf.
[IUTchII],
Corollary
4.8,
(i),
(ii)]
—
are
performed
in
the
context
of
the
log-link,
which
depends,
in
a
quite
essential
way,
on
the
arithmetic
holomorphic
[i.e.,
ring!]
structures
of
the
various
local
fields
involved
—
cf.,
for
instance,
the
discussion
of
the
relevant
log-Kummer
correspondences
in
Remark
3.12.2,
(iv),
(v),
below.
On
the
other
hand,
one
fundamental
difference
between
the
theta
and
number
field
cases
may
be
observed
in
the
fact
that
whereas
j
2
·
the
output
data
in
the
theta
case
—
i.e.,
the
theta
values
“q
”
—
depends,
in
an
essential
way,
on
the
labels
j
∈
F
l
,
v
·
the
output
data
in
the
number
field
case
—
i.e.,
the
copies
of
the
number
field
“F
mod
”
—
is
independent
of
these
labels
j
∈
F
l
.
In
this
context,
let
us
recall
that
these
labels
j
∈
F
l
correspond,
in
essence,
to
collections
of
cuspidal
inertia
groups
[cf.
[IUTchI],
Definition
4.1,
(ii)]
of
the
local
geometric
fundamental
groups
that
appear
[i.e.,
in
the
notation
of
the
discussion
of
Remark
2.2.2,
(iii),
the
subgroup
“Δ
(⊆
Π)”
of
the
local
arithmetic
fundamental
group
Π].
On
the
other
hand,
let
us
recall
that,
in
the
context
of
these
local
arithmetic
fundamental
groups
Π,
the
arithmetic
holomorphic
structure
also
depends,
in
an
essential
way,
on
the
geometric
fundamental
group
portion
[i.e.,
78
SHINICHI
MOCHIZUKI
“Δ
⊆
Π”]
of
Π
[cf.,
e.g.,
the
discussion
of
[AbsTopIII],
Theorem
1.9,
in
[IUTchI],
Remark
3.1.2,
(ii);
the
discussion
of
[AbsTopIII],
§I3].
In
particular,
it
is
a
quite
nontrivial
fact
that
the
Galois
evaluation
and
Kummer
theory
in
the
theta
case
may
be
performed
[cf.
[IUTchII],
Corollary
3.6]
in
a
consistent
fashion
that
is
compatible
with
both
the
labels
j
∈
F
l
[cf.
also
the
associated
sym-
metries
discussed
in
[IUTchII],
Corollary
3.6,
(i)]
and
the
arithmetic
holomorphic
structures
involved
—
i.e.,
both
of
which
depend
on
“Δ”
in
an
essential
way.
By
contrast,
the
corresponding
Galois
evaluation
and
Kummer
theory
operations
in
the
number
field
case
are
performed
[cf.
[IUTchII],
Corollary
4.8,
(i),
(ii)]
in
a
way
that
is
compatible
with
the
arithmetic
holomorphic
structures
involved,
but
yields
output
data
[i.e.,
copies
of
the
number
field
“F
mod
”]
that
is
free
of
any
dependence
on
the
labels
j
∈
F
l
.
Of
course,
the
global
realified
Gaussian
Frobenioids
constructed
in
[IUTchII],
Corollary
4.6,
(v),
which
also
play
an
important
role
in
the
theory
of
the
present
series
of
papers,
involve
global
data
that
depends,
in
an
essential
way,
on
the
labels
j
∈
F
l
,
but
this
dependence
occurs
only
in
the
context
of
global
realified
Frobenioids,
i.e.,
which
[cf.
the
notation
“”
as
it
is
used
in
[IUTchI],
Definition
5.2,
(iv);
[IUTchII],
Definition
4.9,
(viii),
as
well
as
in
Definition
2.4,
(iii),
below]
are
mono-analytic
in
nature
[i.e.,
do
not
depend
on
the
arithmetic
holomorphic
structure
of
copies
of
the
number
field
“F
mod
”].
(iv)
In
the
context
of
the
observations
of
(iii),
we
make
the
further
obser-
vation
that
it
is
a
highly
nontrivial
fact
that
the
construction
algorithm
for
the
mono-theta-theoretic
cyclotomic
rigidity
isomorphism
applied
in
the
theta
case
admits
F
±
l
-symmetries
[cf.
the
discussion
of
[IUTchII],
Remark
1.1.1,
(v);
[IUTchII],
Corollary
3.6,
(i)]
in
a
fashion
that
is
consistent
with
the
dependence
of
the
theta
values
on
the
labels
j
∈
F
l
.
As
discussed
in
[IUTchII],
Remark
1.1.1,
(v),
this
state
of
affairs
differs
quite
substantially
from
the
state
of
affairs
that
arises
in
the
case
of
the
approach
to
cyclotomic
rigidity
taken
in
[IUTchI],
Example
5.1,
(v),
which
is
based
on
a
rather
“straightforward”
or
“naive”
utiliza-
tion
of
the
Kummer
classes
of
rational
functions.
That
is
to
say,
the
“highly
nontrivial”
fact
just
observed
in
the
theta
case
would
amount,
from
the
point
of
view
of
this
“naive
Kummer
approach”
to
cyclotomic
rigidity,
to
the
existence
of
a
rational
function
[or,
alternatively,
a
collection
of
rational
functions
without
“la-
bels”]
that
is
invariant
[up
to,
say,
multiples
by
roots
of
unity]
with
respect
to
the
±
F
±
l
-symmetries
that
appear,
but
nevertheless
attains
values
on
some
F
l
-orbit
of
points
that
have
distinct
valuations
at
distinct
points
—
a
situation
that
is
clearly
self-contradictory!
(v)
One
way
to
appreciate
the
nontriviality
of
the
“highly
nontrivial”
fact
ob-
served
in
(iv)
is
as
follows.
One
possible
approach
to
realizing
the
apparently
“self-
contradictory”
state
of
affairs
constituted
by
a
“symmetric
rational
function
with
non-symmetric
values”
consists
of
replacing
the
local
arithmetic
fundamental
group
“Π”
[cf.
the
notation
of
the
discussion
of
(iii)]
by
some
suitable
closed
subgroup
of
infinite
index
of
Π.
That
is
to
say,
if
one
works
with
such
infinite
index
closed
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
79
subgroups
of
Π,
then
the
possibility
arises
that
the
Kummer
classes
of
those
ratio-
nal
functions
that
constitute
the
obstruction
to
symmetry
in
the
case
of
some
given
rational
function
of
interest
[i.e.,
at
a
more
concrete
level,
the
rational
functions
that
arise
as
quotients
of
the
given
rational
function
by
its
F
±
l
-conjugates]
vanish
upon
restriction
to
such
infinite
index
closed
subgroups
of
Π.
On
the
other
hand,
this
approach
has
the
following
“fundamental
deficiencies”,
both
of
which
relate
to
an
apparently
fatal
lack
of
compatibility
with
the
arithmetic
holomorphic
structures
involved:
·
It
is
not
clear
that
the
absolute
anabelian
results
of
[AbsTopIII],
§1
—
i.e.,
which
play
a
fundamental
role
in
the
theory
of
the
present
series
of
papers
—
admit
generalizations
to
the
case
of
such
infinite
index
closed
subgroups
of
Π.
·
The
vanishing
of
Kummer
classes
of
certain
rational
functions
that
occurs
when
one
restricts
to
such
infinite
index
closed
subgroups
of
Π
will
not,
in
general,
be
compatible
with
the
ring
structures
involved
[i.e.,
of
the
rings/fields
of
rational
functions
that
appear].
In
particular,
this
approach
does
not
appear
to
be
likely
to
give
rise
to
a
meaningful
theory.
(vi)
Another
possible
approach
to
realizing
the
apparently
“self-contradictory”
state
of
affairs
constituted
by
a
“symmetric
rational
function
with
non-symmetric
values”
consists
of
working
with
distinct
rational
functions,
i.e.,
one
symmetric
rational
function
[or
collection
of
rational
functions]
for
constructing
cyclotomic
rigidity
isomorphisms
via
the
Kummer-theoretic
approach
of
[IUTchI],
Example
5.1,
(v),
and
one
non-symmetric
rational
function
to
which
one
applies
Galois
evaluation
operations
to
construct
the
analogue
of
“theta
values”.
On
the
other
hand,
this
approach
has
the
following
“fundamental
deficiency”,
which
again
relates
to
a
sort
of
fatal
lack
of
compatibility
with
the
arithmetic
holomorphic
structures
involved:
The
crucial
absolute
anabelian
results
of
[AbsTopIII],
§1
[cf.
also
the
discussion
of
[IUTchI],
Remark
3.1.2,
(ii),
(iii)],
depend,
in
an
essential
way,
on
the
use
of
numerous
cyclotomes
[i.e.,
copies
of
“
Z(1)”]
—
which,
for
simplicity,
we
shall
denote
by
μ
∗
et
in
the
present
discussion
—
that
arise
from
the
various
cuspidal
inertia
groups
at
the
cusps
“∗”
of
[the
various
cuspidalizations
associated
to]
the
hyperbolic
curve
under
consideration.
These
cyclotomes
“μ
∗
et
”
[i.e.,
for
various
cusps
“∗”]
may
be
naturally
identified
with
one
another,
i.e.,
via
the
natural
isomorphisms
of
[Ab-
sTopIII],
Proposition
1.4,
(ii);
write
μ
∀
et
for
the
cyclotome
resulting
from
this
natural
identification.
Moreover,
since
the
various
[pseudo-]monoids
constructed
by
applying
these
anabelian
results
are
con-
structed
as
sub[-pseudo-]monoids
of
first
[group]
cohomology
modules
with
coeffi-
cients
in
the
cyclotome
μ
∀
et
,
it
follows
[cf.
the
discussion
of
[IUTchII],
Remark
1.5.2]
that
the
cyclotome
μ
Fr
determined
by
[i.e.,
the
cyclotome
obtained
by
applying
Hom(Q/Z,
−)
to
the
tor-
sion
subgroup
of]
such
a
[pseudo-]monoid
may
be
tautologically
identified
—
i.e.,
80
SHINICHI
MOCHIZUKI
whenever
the
[pseudo-]monoid
under
consideration
is
regarded
[not
just
as
an
ab-
stract
“Frobenius-like”
[pseudo-]monoid,
but
rather]
as
the
“étale-like”
output
data
of
an
anabelian
construction
of
the
sort
just
discussed
—
with
the
cyclotome
μ
∀
et
.
In
the
context
of
the
relevant
log-Kummer
correspondences
[i.e.,
as
discussed
in
Remark
3.12.2,
(iv),
(v),
below;
Theorem
3.11,
(ii),
below],
we
shall
work
with
var-
ious
Kummer
isomorphisms
between
such
Frobenius-like
and
étale-like
versions
of
various
[pseudo-]monoids,
i.e.,
in
the
notation
of
the
final
display
of
Proposi-
tion
1.3,
(iv),
between
various
objects
associated
to
the
Frobenius-like
“•’s”
and
corresponding
objects
associated
to
the
étale-like
“◦”.
Now
so
long
as
one
re-
gards
these
various
Frobenius-like
“•’s”
and
the
étale-like
“◦”
as
distinct
labels
for
corresponding
objects,
the
diagram
constituted
by
the
relevant
log-Kummer
correspondence
does
not
result
in
any
“vicious
circles”
or
“loops”.
On
the
other
hand,
ultimately
in
the
theory
of
§3
[cf.,
especially,
the
final
portion
of
Theorem
3.11,
(iii),
(c),
(d),
below;
the
proof
of
Corollary
3.12
below],
we
shall
be
interested
in
applying
the
theory
to
the
task
of
constructing
algorithms
to
describe
objects
of
interest
of
one
arithmetic
holomorphic
structure
in
terms
of
some
alien
arithmetic
holomorphic
structure
[cf.
Remark
3.11.1]
by
means
of
“multiradial
containers”
[cf.
Remark
3.12.2,
(ii)].
These
multiradial
containers
arise
from
étale-like
versions
of
objects,
but
are
ultimately
applied
as
containers
for
Frobenius-like
versions
of
objects.
That
is
to
say,
in
order
for
such
multiradial
containers
to
function
as
containers,
it
is
necessary
to
contend
with
the
consequences
of
identifying
the
Frobenius-
like
and
étale-like
versions
of
various
objects
under
consideration,
e.g.,
in
the
context
of
the
above
discussion,
of
identifying
μ
Fr
with
μ
∀
et
.
On
the
other
hand,
let
us
recall
that
the
approach
to
constructing
cyclotomic
rigid-
ity
isomorphisms
associated
to
rational
functions
via
the
Kummer-theoretic
ap-
proach
of
[IUTchI],
Example
5.1,
(v),
amounts
in
effect
[i.e.,
in
the
context
of
the
above
discussion],
to
“identifying”
various
“μ
∗
et
’s”
with
various
“sub-cyclotomes”
of
“μ
Fr
”
via
morphisms
that
differ
from
the
usual
natural
identification
precisely
by
multiplication
by
the
order
[∈
Z]
at
“∗”
of
the
zeroes/poles
of
the
rational
function
under
consideration.
That
is
to
say,
to
execute
such
a
cyclotomic
rigidity
isomorphism
construction
in
a
situation
subject
to
the
further
identification
of
μ
Fr
with
μ
∀
et
[which,
we
recall,
was
obtained
by
identifying
the
various
“μ
∗
et
’s”!]
does
indeed
result
—
at
least
in
an
a
priori
sense!
—
in
“vicious
circles”/“loops”
[cf.
the
discussion
of
[IUTchIV],
Remark
3.3.1,
(i);
the
reference
to
this
discussion
in
[IUTchI],
Remark
4.3.1,
(ii)].
That
is
to
say,
in
order
to
avoid
any
possible
contradictions
that
might
arise
from
such
“vicious
circles”/“loops”,
it
is
necessary
to
work
with
objects
that
are
“invariant”,
or
“coric”,
with
respect
to
such
“vicious
circles”/“loops”,
i.e.,
to
regard
the
cyclotome
μ
∀
et
as
being
subject
to
indeterminacies
with
respect
to
multiplication
by
elements
of
the
submonoid
def
I
ord
⊆
±N
≥1
=
N
≥1
×
{±1}
generated
by
the
orders
[∈
Z]
of
the
zeroes/poles
of
the
rational
func-
tion(s)
that
appear
in
the
cyclotomic
rigidity
isomorphism
construction
under
consideration.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
81
ord
In
the
following
discussion,
we
shall
also
write
I
ord
≥1
⊆
N
≥1
,
I
±
⊆
{±1}
for
the
respective
images
of
I
ord
via
the
natural
projections
to
N
≥1
,
{±1}.
This
sort
of
indeterminacy
is
fundamentally
incompatible,
for
numerous
reasons,
with
any
sort
of
construction
that
purports
to
be
analogous
to
the
construction
of
the
“theta
values”
in
the
theory
of
the
present
series
of
papers,
i.e.,
at
least
whenever
the
resulting
indeterminacy
submonoid
I
ord
⊆
±N
≥1
is
nontrivial.
For
instance,
it
follows
immediately,
by
considering
the
effect
of
independent
indeterminacies
of
this
type
on
valuations
at
distinct
v
∈
V,
that
such
independent
indeterminacies
are
incompatible
with
the
“product
formula”
[i.e.,
with
the
structure
of
the
global
realified
Frobenioids
involved
—
cf.
[IUTchI],
Remark
3.5.1,
(ii)].
Here,
we
observe
that
this
sort
of
indeterminacy
does
not
occur
in
the
theta
case
[cf.
Fig.
2.5
below]
—
i.e.,
the
resulting
indeterminacy
submonoid
(±N
≥1
⊇)
I
ord
=
{1}
—
precisely
as
a
consequence
of
the
fact
[which
is
closely
related
to
the
symmetry
properties
discussed
in
[IUTchII],
Remark
1.1.1,
(v)]
that
the
order
[∈
Z]
of
the
zeroes/poles
of
the
theta
function
at
every
cusp
is
equal
to
1
[cf.
[EtTh],
Proposition
1.4,
(i);
[IUTchI],
Remark
3.1.2,
(ii),
(iii)]
—
a
state
of
affairs
that
can
never
occur
in
the
case
of
an
algebraic
rational
function
[i.e.,
since
the
sum
of
the
orders
[∈
Z]
of
the
zeroes/poles
of
an
algebraic
rational
function
is
always
equal
to
0]!
On
the
other
hand,
in
the
number
field
case
[cf.
Fig.
2.6
below],
the
portion
of
the
indeterminacy
under
consideration
that
is
constituted
by
I
ord
≥1
is
avoided
precisely
[cf.
the
discussion
of
[IUTchI],
Example
5.1,
(v)]
by
applying
the
property
×
=
{1}
Z
Q
>0
×
-
[cf.
also
the
discussion
of
(vii)
below!],
which
has
the
effect
of
isolating
the
Z
torsor
of
interest
[i.e.,
some
specific
isomorphism
between
cyclotomes]
from
the
subgroup
of
Q
>0
generated
by
I
ord
≥1
.
This
technique
for
avoiding
the
indeterminacy
ord
constituted
by
I
≥1
remains
valid
even
after
the
identification
discussed
above
of
μ
Fr
with
μ
∀
et
.
By
contrast,
the
portion
of
the
indeterminacy
under
consideration
that
is
constituted
by
I
ord
±
is
avoided
in
the
construction
of
[IUTchI],
Example
5.1,
(v),
precisely
by
applying
the
fact
that
the
inverse
of
a
nonconstant
κ-coric
rational
function
is
never
κ-coric
[cf.
the
discussion
of
[IUTchI],
Remark
3.1.7,
(i)]
—
a
technique
that
depends,
in
an
essential
way,
on
distinguishing
cusps
“∗”
at
which
the
orders
[∈
Z]
of
the
zeroes/poles
of
the
rational
function(s)
under
consideration
are
distinct.
In
particular,
this
technique
is
fundamentally
incompatible
with
the
identification
discussed
above
of
μ
Fr
with
μ
∀
et
.
That
is
to
say,
in
summary,
in
the
number
field
case,
in
order
to
regard
étale-like
versions
of
objects
as
containers
for
Frobenius-like
versions
of
objects,
it
is
necessary
to
regard
the
relevant
cyclotomic
rigidity
isomorphisms
—
hence
also
the
output
data
of
interest
in
the
number
field
case,
i.e.,
copies
of
×
”
—
as
being
subject
to
an
[the
union
with
{0}
of]
the
group
“F
mod
indeterminacy
constituted
by
[possible]
multiplication
by
{±1}.
This
does
not
result
in
any
additional
technical
obstacles,
however,
since
82
SHINICHI
MOCHIZUKI
the
output
data
of
interest
in
the
number
field
case
—
i.e.,
copies
of
×
”
—
is
[unlike
the
case
with
the
[the
union
with
{0}
of]
the
group
“F
mod
j
2
theta
values
“q
”!]
stabilized
by
the
action
of
{±1}
v
—
cf.
the
discussion
of
Remark
3.11.4
below.
Moreover,
we
observe
in
passing,
in
the
context
of
the
Galois
evaluation
operations
in
the
number
field
case,
that
the
×
”
are
constructed
globally
and
in
a
fashion
compatible
copies
of
[the
group]
“F
mod
with
the
F
-symmetry
[cf.
[IUTchII],
Corollary
4.8,
(i),
(ii)],
hence,
in
particular,
l
in
a
fashion
that
does
not
require
the
establishment
of
compatibility
properties
[e.g.,
relating
to
the
“product
formula”]
between
constructions
at
distinct
v
∈
V.
...
+1
∗
+1
∗
+1
∗
+1
∗
...
+1
∗
+1
∗
+1
∗
+1
∗
...
Fig.
2.5:
Orders
[∈
Z]
of
zeroes/poles
of
the
theta
function
at
the
cusps
“∗”
...
0
∗
0
∗
+8
∗
−5
∗
...
−6
∗
+3
∗
0
∗
0
∗
...
Fig.
2.6:
Orders
[∈
Z]
of
zeroes/poles
of
an
algebraic
rational
function
at
the
cusps
“∗”
(vii)
In
the
context
of
the
discussion
of
(vi),
we
observe
that
the
indeterminacy
issues
discussed
in
(vi)
may
be
thought
of
as
a
sort
of
“multiple
cusp
version”
of
the
“N
-th
power
versus
first
power”
and
“linearity”
issues
discussed
in
[IUTchII],
Remark
3.6.4,
(iii).
Also,
in
this
context,
we
recall
from
the
discussion
at
the
beginning
of
Remark
2.1.1
that
the
theory
of
mono-theta-theoretic
cy-
clotomic
rigidity
satisfies
the
important
property
of
being
compatible
with
the
topology
of
the
tempered
fundamental
group.
Such
a
compatibility
contrasts
sharply
with
the
cyclotomic
rigidity
algorithms
discussed
in
[IUTchI],
Example
5.1,
(v),
which
depend
[cf.
the
discussion
of
(vi)
above!],
in
an
essential
way,
on
the
property
×
=
{1}
Z
Q
>0
—
i.e.,
which
is
fundamentally
incompatible
with
the
topology
of
the
profinite
groups
involved
[as
can
be
seen,
for
instance,
by
considering
the
fact
that
N
≥1
forms
This
close
relationship
between
cyclotomic
rigidity
and
[a
a
dense
subset
of
Z].
sort
of]
discrete
rigidity
[i.e.,
the
property
of
the
above
display]
is
reminiscent
of
the
discussion
given
in
[IUTchII],
Remark
2.8.3,
(ii),
of
such
a
relationship
in
the
case
of
mono-theta
environments.
(viii)
In
the
context
of
the
discussion
of
(vi),
(vii),
we
observe
that
the
inde-
terminacy
issues
discussed
in
(vi)
also
occur
in
the
case
of
the
cyclotomic
rigidity
algorithms
discussed
in
[IUTchI],
Definition
5.2,
(vi),
i.e.,
in
the
context
of
mixed-
characteristic
local
fields.
On
the
other
hand,
[cf.
[IUTchII],
Proposition
4.2,
(i)]
these
algorithms
in
fact
yield
the
same
cyclotomic
rigidity
isomorphism
as
the
cy-
clotomic
rigidity
isomorphisms
that
are
applied
in
[AbsTopIII],
Proposition
3.2,
(iv)
[i.e.,
the
cyclotomic
rigidity
isomorphisms
discussed
in
[AbsTopIII],
Proposition
3.2,
(i),
(ii);
[AbsTopIII],
Remark
3.2.1].
Moreover,
these
cyclotomic
rigidity
isomor-
phisms
discussed
in
[AbsTopIII]
are
manifestly
compatible
with
the
topology
of
the
profinite
groups
involved.
From
the
point
of
view
of
the
discussion
of
(vi),
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
83
this
sort
of
“de
facto”
compatibility
with
the
topology
of
the
profinite
groups
involved
may
be
thought
of
as
a
reflection
of
the
fact
that
these
cyclotomic
rigidity
isomor-
phisms
discussed
in
[AbsTopIII]
amount,
in
essence,
to
applying
the
approach
to
cyclotomic
rigidity
by
considering
the
Kummer
theory
of
algebraic
rational
func-
tions
[i.e.,
the
approach
of
(vi),
or,
alternatively,
of
[IUTchI],
Example
5.1,
(v)],
in
the
case
where
the
algebraic
rational
functions
are
taken
to
be
the
uniformizers
—
i.e.,
“rational
functions”
[any
one
of
which
is
well-defined
up
to
a
unit]
with
pre-
cisely
one
zero
of
order
1
and
no
poles
[cf.
the
discussion
of
the
theta
function
in
(vi)!]
—
of
the
mixed-characteristic
local
field
under
consideration.
Put
another
way,
this
sort
of
“de
facto”
compatibility
may
be
regarded
as
a
reflection
of
the
fact
that,
unlike
number
fields
[i.e.,
“NF’s”]
or
one-dimensional
function
fields
[i.e.,
“one-dim.
FF’s”],
mixed-characteristic
local
fields
[i.e.,
“MLF’s”]
are
equipped
with
a
uniquely
determined
“canonical
valuation”
—
a
situation
that
is
rem-
iniscent
of
the
fact
that
the
order
[∈
Z]
of
the
zeroes/poles
of
the
theta
function
at
every
cusp
is
equal
to
1
[i.e.,
the
fact
that
“the
set
of
equivalences
classes
of
cusps
relative
to
the
equivalence
relationship
on
cusps
determined
by
considering
the
or-
der
[∈
Z]
of
the
zeroes/poles
of
the
theta
function
is
of
cardinality
one”].
From
the
point
of
view
of
“geometric
containers”
discussed
in
(i)
and
(ii),
this
state
of
affairs
may
be
summarized
as
follows:
the
indeterminacy
issues
that
occur
in
the
context
of
the
discussion
of
cyclotomic
rigidity
isomorphisms
in
(vi)
exhibit
similar
qualitative
behavior
in
the
MLF/mono-theta
(←→
one
valuation/cusp)
[i.e.,
where
the
expression
“one
cusp”
is
to
be
understood
as
referring
to
“one
equivalence
class
of
cusps”,
as
discussed
above]
cases,
as
well
as
in
the
NF/one-dim.
FF
(←→
global
collection
of
valuations/cusps)
cases.
Put
another
way,
at
least
at
the
level
of
the
theory
of
valuations,
the
theory
of
theta
functions
(respectively,
one-dimensional
function
fields)
serves
as
an
accurate
“qualitative
geometric
model”
of
the
the-
ory
of
mixed-characteristic
local
fields
(respectively,
number
fields).
×
=
{1}”
Z
Finally,
we
observe
that
in
this
context,
the
crucial
property
“Q
>0
that
occurs
in
the
discussion
of
the
number
field/one-dimensional
function
field
cases
is
highly
reminiscent
of
the
global
nature
of
number
fields
[i.e.,
such
as
Q!
—
cf.
the
discussion
of
Remark
3.12.1,
(iii),
below].
(ix)
The
comparison
given
in
(viii)
of
the
special
properties
satisfied
by
the
theta
function
with
the
corresponding
properties
of
the
algebraic
rational
func-
tions
that
appear
in
the
number
field
case
is
reminiscent
of
the
analogy
discussed
in
[IUTchI],
Remark
6.12.3,
(iii),
with
the
classical
upper
half-plane.
That
is
to
say,
the
eigenfunction
for
the
additive
symmetries
of
the
upper
half-plane
[i.e.,
which
corresponds
to
the
theta
case]
q
def
=
e
2πiz
84
SHINICHI
MOCHIZUKI
Aspect
of
the
theory
Theta
case
Number
field
case
multiradial
geometric
container
theta
functions
on
Tate
curves
Belyi
maps/
cuspidalizations
radial
arithmetic
theta
values
copies
of
j
2
data
via
“q
”
v
number
field
“F
mod
”
×
⊇
F
mod
{±1}
Galois
evaluation
Galois
evaluation
output
data
dependence
on
“Δ”
simultaneously
dependent
on
labels,
holomorphic
str.
indep.
of
labels,
dependent
on
holomorphic
str.
cyclotomic
rigidity
isomorphism
compatible
with
F
±
l
-symmetry,
tempered
topology
incompatible
with
F
±
l
-symmetry,
profinite
topology
approach
to
eliminating
cyclo.
rig.
isom.
indeterminacies
order
[∈
Z]
of
zeroes/poles
of
theta
function
at
every
cusp
=
1
×
=
{1},
Z
Q
>0
non-invertibility
of
nonconstant
κ-coric
rational
functions
qualitative
geometric
model
for
arithmetic
MLF/mono-theta
(←→
one
valuation/cusp)
analogy
NF/one-dim.
FF
(←→
global
collection
of
valuations/cusps)
analogy
analogy
with
eigenfunctions
for
symmetries
of
highly
transcendental
function
in
z:
algebraic
rational
function
of
z:
upper
half-plane
q
def
=
e
2πiz
w
def
=
z−i
z+i
Fig.
2.7:
Comparison
between
the
theta
and
number
field
cases
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
85
is
highly
transcendental
in
the
coordinate
z,
whereas
the
eigenfunction
for
the
multiplicative
symmetries
of
the
upper
half-plane
[i.e.,
which
corresponds
to
the
number
field
case]
w
def
=
z−i
z+i
is
an
algebraic
rational
function
in
the
coordinate
z.
(x)
The
various
properties
discussed
above
in
the
theta
and
number
field
cases
are
summarized
in
Fig.
2.7
above.
Remark
2.3.4.
Before
proceeding,
it
is
perhaps
of
interest
to
review
once
more
the
essential
content
of
[EtTh]
in
light
of
the
various
observations
made
in
Remark
2.3.3.
(i)
The
starting
point
of
the
relationship
between
the
theory
of
[EtTh]
and
the
theory
of
the
present
series
of
papers
lies
[cf.
the
discussion
of
Remark
2.1.1,
(i);
[IUTchII],
Remark
3.6.2,
(ii)]
in
the
various
non-ring/scheme-theoretic
filters
[i.e.,
log-links
and
various
types
of
Θ-links]
between
distinct
ring/scheme
theories
that
are
constructed
in
the
present
series
of
papers.
Such
non-scheme-theoretic
fil-
ters
may
only
be
constructed
by
making
use
of
Frobenius-like
structures.
On
the
other
hand,
étale-like
structures
are
important
in
light
of
their
ability
to
relate
structures
on
opposite
sides
of
such
non-scheme-theoretic
filters.
Then
Kummer
theory
is
applied
to
relate
corresponding
Frobenius-like
and
étale-like
structures.
Moreover,
it
is
crucial
that
this
Kummer
theory
be
conducted
in
a
multiradial
fashion.
This
is
achieved
by
means
of
certain
radial/coric
decouplings,
by
mak-
ing
use
of
multiradial
geometric
containers,
as
discussed
in
Remark
2.3.3,
(i),
(ii).
That
is
to
say,
it
is
necessary
to
make
use
of
such
multiradial
geometric
containers
and
then
to
pass
to
theta
values
or
number
fields
by
means
of
Galois
evaluation,
since
direct
use
of
such
theta
values
or
number
fields
results
in
a
Kum-
mer
theory
that
does
not
satisfy
the
desired
multiradiality
properties
[cf.
Remarks
2.2.1,
2.3.2].
(ii)
The
most
naive
approach
to
the
Kummer
theory
of
the
“functions”
that
are
to
be
used
as
“multiradial
geometric
containers”
may
be
seen
in
the
approach
involving
algebraic
rational
functions
on
the
various
algebraic
curves
under
consideration,
i.e.,
in
the
fashion
of
[IUTchI],
Example
5.1,
(v)
[cf.
also
[IUTchI],
Definition
5.2,
(vi)].
On
the
other
hand,
in
the
context
of
the
local
theory
at
v
∈
V
bad
,
this
approach
suffers
from
the
fatal
drawback
of
being
incompatible
with
the
profinite
topology
of
the
profinite
fundamental
groups
involved
[cf.
the
discussion
of
Remark
2.3.3,
(vi),
(vii),
(viii);
Figs.
2.5,
2.6].
Thus,
in
order
to
main-
tain
compatibility
with
the
profinite/tempered
topology
of
the
profinite/tempered
fundamental
groups
involved,
one
is
obliged
to
work
with
the
Kummer
theory
of
theta
functions,
truncated
modulo
N
.
On
the
other
hand,
the
naive
ap-
proach
to
this
sort
of
[truncated
modulo
N
]
Kummer
theory
of
theta
functions
suffers
from
the
fatal
drawback
of
being
incompatible
with
discrete
rigidity
[cf.
Remark
2.1.1,
(v)].
This
incompatibility
with
discrete
rigidity
arises
from
a
lack
of
“shifting
automorphisms”
as
in
[EtTh],
Proposition
2.14,
(ii)
[cf.
also
[EtTh],
Remark
2.14.3],
and
is
closely
related
to
the
incompatibility
of
this
naive
ap-
proach
with
the
F
±
l
-symmetry
[cf.
the
discussion
of
[IUTchII],
Remark
1.1.1,
86
SHINICHI
MOCHIZUKI
(iv),
(v)].
In
order
to
surmount
such
incompatibilities,
one
is
obliged
to
consider
not
the
Kummer
theory
of
theta
functions
in
the
naive
sense,
but
rather,
so
to
speak,
the
Kummer
theory
of
[the
first
Chern
classes
of]
the
line
bundles
associated
to
theta
functions
[cf.
the
discussion
of
[IUTchII],
Remark
1.1.1,
(v)].
Thus,
in
summary:
[truncated]
Kummer
theory
of
theta
[not
algebraic
rational!]
functions
=⇒
compatible
with
profinite/tempered
topologies;
[truncated]
Kummer
theory
of
[first
Chern
classes
of]
line
bundles
[not
rational
functions!]
=⇒
compatible
with
discrete
rigidity,
F
±
l
-symmetry.
(iii)
To
consider
the
“[truncated]
Kummer
theory
of
line
bundles
[associated
to
the
theta
function]”
amounts,
in
effect,
to
considering
the
[partially
truncated]
arithmetic
fundamental
group
of
the
G
m
-torsor
determined
by
such
a
line
bundle
in
a
fashion
that
is
compatible
with
the
various
tempered
Frobenioids
and
tem-
pered
fundamental
groups
under
consideration.
Such
a
“[partially
truncated]
arithmetic
fundamental
group”
corresponds
precisely
to
the
“topological
group”
por-
tion
of
the
data
that
constitutes
a
mono-theta
or
bi-theta
environment
[cf.
[EtTh],
Definition
2.13,
(ii),
(a);
[EtTh],
Definition
2.13,
(iii),
(a)].
In
the
context
of
the
the-
ory
of
theta
functions,
such
“[partially
truncated]
arithmetic
fundamental
groups”
are
equipped
with
two
natural
distinguished
[classes
of
]
sections,
namely,
theta
sections
and
algebraic
sections.
If
one
thinks
of
the
[partially
truncated]
arith-
metic
fundamental
groups
under
consideration
as
being
equipped
neither
with
data
corresponding
to
theta
sections
nor
with
data
corresponding
to
algebraic
sections,
then
the
resulting
mathematical
object
is
necessarily
subject
to
indeterminacies
arising
from
multiplication
by
constant
units
[i.e.,
“O
×
”
of
the
base
local
field],
hence,
in
particular,
suffers
from
the
drawback
of
being
incompatible
with
constant
multiple
rigidity
[cf.
Remark
2.1.1,
(iii)].
On
the
other
hand,
if
one
thinks
of
the
[partially
truncated]
arithmetic
fundamental
groups
under
consideration
as
being
equipped
both
with
data
corresponding
to
theta
sections
and
with
data
correspond-
ing
to
algebraic
sections,
then
the
resulting
mathematical
object
suffers
from
the
same
lack
of
symmetries
as
the
[truncated]
Kummer
theory
of
theta
functions
[cf.
the
discussion
of
(ii)],
hence,
in
particular,
is
incompatible
with
discrete
rigidity
[cf.
Remark
2.1.1,
(v)].
Finally,
if
one
thinks
of
the
[partially
truncated]
arith-
metic
fundamental
groups
under
consideration
as
being
equipped
only
with
data
corresponding
to
algebraic
sections
[i.e.,
but
not
with
data
corresponding
to
theta
sections!],
then
the
resulting
mathematical
object
is
not
equipped
with
sufficient
data
to
apply
the
crucial
commutator
property
of
[EtTh],
Proposition
2.12
[cf.
also
the
discussion
of
[EtTh],
Remark
2.19.2],
hence,
in
particular,
is
incompatible
with
cyclotomic
rigidity
[cf.
Remark
2.1.1,
(iv)].
That
is
to
say,
it
is
only
by
thinking
of
the
[partially
truncated]
arithmetic
fundamental
groups
under
consideration
as
being
equipped
only
with
data
corresponding
to
theta
sections
[i.e.,
but
not
with
data
corresponding
to
algebraic
sections!]
—
i.e.,
in
short,
by
working
with
mono-
theta
environments
—
that
one
may
achieve
a
situation
that
is
compatible
with
the
tempered
topology
of
the
tempered
fundamental
groups
involved,
the
F
±
l
-symmetry,
and
all
three
types
of
rigidity
[cf.
the
initial
portion
of
Remark
2.1.1;
[IUTchII],
Remark
3.6.4,
(ii)].
Thus,
in
summary:
working
neither
with
theta
sections
nor
with
algebraic
sections
=⇒
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
87
incompatible
with
constant
multiple
rigidity!
working
with
bi-theta
environments,
i.e.,
working
simultaneously
with
both
theta
sections
and
algebraic
sections
=⇒
incompatible
with
discrete
rigidity,
F
±
l
-symmetry!
working
with
algebraic
sections
but
not
theta
sections
=⇒
incompatible
with
cyclotomic
rigidity!
working
with
mono-theta
environments,
i.e.,
working
with
theta
sections
but
not
algebraic
sections
=⇒
compatible
with
tempered
topology,
F
±
l
-symmetry,
all
three
rigidities!
(iv)
Finally,
we
note
that
the
approach
of
[EtTh]
to
the
theory
of
theta
func-
tions
differs
substantially
from
more
conventional
approaches
to
the
theory
of
theta
functions
such
as
·
the
classical
function-theoretic
approach
via
explicit
series
repre-
sentations,
i.e.,
as
given
at
the
beginning
of
the
Introduction
to
[IUTchII]
[cf.
also
[EtTh],
Proposition
1.4],
and
·
the
representation-theoretic
approach,
i.e.,
by
considering
irreducible
representations
of
theta
groups.
Both
of
these
more
conventional
approaches
depend,
in
an
essential
way,
on
the
ring
structures
—
i.e.,
on
both
the
additive
and
the
multiplicative
structures
—
of
the
various
rings
involved.
[Here,
we
recall
that
explicit
series
are
constructed
precisely
by
adding
and
multiplying
various
functions
on
some
space,
whereas
rep-
resentations
are,
in
effect,
modules
over
suitable
rings,
hence,
by
definition,
involve
both
additive
and
multiplicative
structures.]
In
particular,
although
these
more
conventional
approaches
are
well-suited
to
many
situations
in
which
one
considers
“the”
theta
function
in
some
fixed
model
of
scheme/ring
theory,
they
are
ill-suited
to
the
situations
treated
in
the
present
series
of
papers,
i.e.,
where
one
must
consider
theta
functions
that
appear
in
various
distinct
ring/scheme
theories,
which
[cf.
the
discussion
of
(i)]
may
only
be
related
to
one
another
by
means
of
suitable
Frobenius-like
and
étale-like
structures
such
as
tempered
Frobenioids
and
tem-
pered
fundamental
groups.
Here,
we
recall
that
these
tempered
Frobenioids
cor-
respond
essentially
to
multiplicative
monoid
structures
arising
from
the
various
rings
of
functions
that
appear,
whereas
tempered
fundamental
groups
correspond
to
various
Galois
actions.
That
is
to
say,
consideration
of
such
multiplicative
monoid
structures
and
Galois
actions
is
compatible
with
the
dismantling
of
the
additive
and
multiplicative
structures
of
a
ring,
i.e.,
as
considered
in
the
present
series
of
papers
[cf.
the
discussion
of
Remark
3.12.2
below].
Definition
2.4.
(i)
Let
‡
F
=
{
‡
F
v
}
v∈V
88
SHINICHI
MOCHIZUKI
be
an
F
-prime-strip.
Then
recall
from
the
discussion
of
[IUTchII],
Definition
4.9,
(ii),
that
at
each
w
∈
V
bad
,
the
splittings
of
the
split
Frobenioid
‡
F
w
determine
submonoids
“O
⊥
(−)
⊆
O
(−)”,
as
well
as
quotient
monoids
“O
⊥
(−)
O
(−)”
[i.e.,
by
forming
the
quotient
of
“O
⊥
(−)”
by
its
torsion
subgroup].
In
a
similar
vein,
for
each
w
∈
V
good
,
the
splitting
of
the
split
Frobenioid
determined
by
[indeed,
“constituted
by”,
when
w
∈
V
good
V
non
—
cf.
[IUTchI],
Definition
5.2,
(ii)]
‡
F
w
determines
a
submonoid
“O
⊥
(−)
⊆
O
(−)”
whose
subgroup
of
units
is
trivial
[cf.
[IUTchII],
Definition
4.9,
(iv),
when
w
∈
V
good
V
non
];
in
this
case,
we
set
def
O
(−)
=
O
⊥
(−).
Write
‡
⊥
F
=
{
‡
F
v
⊥
}
v∈V
;
‡
F
=
{
‡
F
v
}
v∈V
for
the
collections
of
data
obtained
by
replacing
the
split
Frobenioid
portion
of
each
‡
F
v
by
the
Frobenioids
determined,
respectively,
by
the
subquotient
monoids
“O
⊥
(−)
⊆
O
(−)”,
“O
(−)”
just
defined.
(ii)
We
define
[in
the
spirit
of
[IUTchII],
Definition
4.9,
(vii)]
an
F
⊥
-prime-
strip
to
be
a
collection
of
data
∗
⊥
F
=
{
∗
F
v
⊥
}
v∈V
that
satisfies
the
following
conditions:
(a)
if
v
∈
V
non
,
then
∗
F
v
⊥
is
a
Frobenioid
that
is
isomorphic
to
‡
F
v
⊥
[cf.
(i)];
(b)
if
v
∈
V
arc
,
then
∗
F
v
⊥
consists
of
a
Frobenioid
and
an
object
of
TM
[cf.
[IUTchI],
Definition
5.2,
(ii)]
such
that
∗
F
v
⊥
is
isomorphic
to
‡
F
v
⊥
.
In
a
similar
vein,
we
define
an
F
-prime-strip
to
be
a
collection
of
data
∗
F
=
{
∗
F
v
}
v∈V
that
satisfies
the
following
conditions:
(a)
if
v
∈
V
non
,
then
∗
F
v
is
a
Frobenioid
that
is
isomorphic
to
‡
F
v
[cf.
(i)];
(b)
if
v
∈
V
arc
,
then
∗
F
v
consists
of
a
Frobenioid
and
an
object
of
TM
[cf.
[IUTchI],
Definition
5.2,
(ii)]
such
that
∗
F
v
is
isomorphic
to
‡
F
v
.
A
morphism
of
F
⊥
-
(respectively,
F
-)
prime-strips
is
defined
to
be
a
collection
of
isomorphisms,
indexed
by
V,
between
the
various
constituent
objects
of
the
prime-strips
[cf.
[IUTchI],
Definition
5.2,
(iii)].
(iii)
We
define
[in
the
spirit
of
[IUTchII],
Definition
4.9,
(viii)]
an
F
⊥
-prime-
strip
to
be
a
collection
of
data
∗
⊥
F
∼
=
(
∗
C
,
Prime(
∗
C
)
→
V,
∗
F
⊥
,
{
∗
ρ
v
}
v∈V
)
satisfying
the
conditions
(a),
(b),
(c),
(d),
(e),
(f)
of
[IUTchI],
Definition
5.2,
(iv),
for
an
F
-prime-strip,
except
that
the
portion
of
the
collection
of
data
constituted
by
an
F
-prime-strip
is
replaced
by
an
F
⊥
-prime-strip.
[We
leave
the
routine
details
to
the
reader.]
In
a
similar
vein,
we
define
an
F
-prime-strip
to
be
a
collection
of
data
∗
F
∼
=
(
∗
C
,
Prime(
∗
C
)
→
V,
∗
F
,
{
∗
ρ
v
}
v∈V
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
89
satisfying
the
conditions
(a),
(b),
(c),
(d),
(e),
(f)
of
[IUTchI],
Definition
5.2,
(iv),
for
an
F
-prime-strip,
except
that
the
portion
of
the
collection
of
data
constituted
by
an
F
-prime-strip
is
replaced
by
an
F
-prime-strip.
[We
leave
the
routine
details
to
the
reader.]
A
morphism
of
F
⊥
-
(respectively,
F
-)
prime-strips
is
defined
to
be
an
isomorphism
between
collections
of
data
as
discussed
above.
Remark
2.4.1.
(i)
Thus,
by
applying
the
constructions
of
Definition
2.4,
(i),
to
the
[underlying
†
F
-prime-strips
associated
to
the]
F
-prime-strips
“F
env
(
D
>
)”
that
appear
in
Corollary
2.3,
one
may
regard
the
multiradiality
of
Corollary
2.3,
(i),
as
implying
a
corresponding
multiradiality
assertion
concerning
the
associated
F
⊥
-prime-
†
strips
“F
⊥
env
(
D
>
)”.
(ii)
Suppose
that
we
are
in
the
situation
discussed
in
(i).
Then
at
v
∈
V
bad
,
the
submonoids
“O
⊥
(−)
⊆
O
(−)”
may
be
regarded,
in
a
natural
way
[cf.
Proposition
†
2.1,
(ii);
Theorem
2.2,
(ii)],
as
submonoids
of
the
monoids
“
∞
Ψ
⊥
env
(
D
>
)
v
”
of
The-
orem
2.2,
(ii),
(a
v
).
Moreover,
the
resulting
inclusion
of
monoids
is
compatible
with
the
multiradiality
discussed
in
(i)
and
the
multiradiality
of
the
data
“
†
R
bad
”
of
Corollary
2.3,
(c
R
),
that
is
implied
by
the
multiradiality
of
Corollary
2.3,
(i).
Remark
2.4.2.
(i)
One
verifies
immediately
that,
just
as
one
may
associate
to
an
F
×μ
-
prime-strip
a
pilot
object
in
the
global
realified
Frobenioid
portion
of
the
F
×μ
-
prime-strip
[cf.
[IUTchII],
Definition
4.9,
(viii)],
one
may
associate
to
an
F
-
prime-strip
a
pilot
object
in
the
global
realified
Frobenioid
portion
of
the
F
-
prime-strip
[i.e.,
in
the
notation
of
the
final
display
of
Definition
2.4,
(iii),
the
global
realified
Frobenioid
∗
C
of
the
F
-prime-strip
∗
F
].
def
(ii)
For
v
∈
V
lying
over
v
∈
V
mod
and
v
Q
∈
V
Q
=
V(Q),
write
def
·
r
v
=
[(F
mod
)
v
:
Q
v
Q
]
·
log(p
v
)
∈
R
if
v
∈
V
good
,
def
·
r
v
=
[(F
mod
)
v
:
Q
v
Q
]
·
ord
v
(q
)
·
log(p
v
)
∈
R
if
v
∈
V
bad
v
—
where,
if
v
∈
V
bad
,
then
ord
v
:
K
v
×
→
Q
denotes
the
natural
p
v
-adic
valuation
normalized
so
that
ord
v
(p
v
)
=
1,
and
q
is
as
in
[IUTchI],
Example
3.2,
(iv);
v
r
v
=
−
def
r
v
r
w
w∈V
bad
[cf.
the
constructions
of
[IUTchI],
Example
3.5;
[IUTchI],
Remark
3.5.1;
the
dis-
cussion
of
weights
in
Remark
3.1.1,
(ii),
below].
(iii)
In
the
notation
of
(ii),
let
M
be
any
ordered
monoid
isomorphic
[as
an
ordered
monoid]
to
R
[endowed
with
the
usual
additive
and
order
structures].
Then
M
naturally
determines
a
collection
of
data
∼
(M,
{M
v
}
v∈V
,
{ρ
M
v
:
M
v
→
M
}
v∈V
)
90
SHINICHI
MOCHIZUKI
∼
as
follows:
for
each
v
∈
V,
we
take
M
v
to
be
a
copy
of
M
and
ρ
M
v
:
M
v
→
M
to
be
the
isomorphism
of
monoids
[that
reverses
the
ordering!]
given
by
multiplying
by
r
v
∈
R.
(iv)
In
the
notation
of
(ii),
(iii),
suppose,
further,
that
we
have
been
a
negative
element
η
M
∈
M
[i.e.,
an
element
<
0],
which
we
shall
refer
to
as
a
pilot
element.
Then,
since,
for
v
∈
V,
M
v
is
defined
to
be
a
copy
of
M
,
η
M
determines
an
element
η
M
v
∈
M
v
.
Thus,
the
pair
(M,
η
M
)
naturally
determines
a
collection
of
data
(M,
{M
v
}
v∈V
,
{ρ
M
v
:
M
v
→
M
}
v∈V
)
as
follows:
for
each
v
∈
V
non
,
we
take
M
v
⊆
M
v
to
be
the
submonoid
[isomorphic
to
N]
generated
by
η
M
v
and
ρ
M
v
:
M
v
→
M
to
be
the
restriction
of
ρ
M
v
to
M
v
;
for
each
v
∈
V
arc
,
we
take
M
v
⊆
M
v
to
be
the
submonoid
[isomorphic
to
R
≥0
]
given
by
the
elements
≤
0
and
ρ
M
v
:
M
v
→
M
to
be
the
restriction
of
ρ
M
v
to
M
v
.
In
particular,
it
follows
immediately
from
the
construction
of
this
data
that
ρ
M
v
(η
M
v
)
=
r
v
·
η
M
for
each
v
∈
V.
(v)
Now
we
observe
that
the
constructions
of
(iii)
and
(iv)
allow
one
to
give
a
sort
of
“converse”
to
the
construction
of
the
pilot
object
in
(i).
Indeed,
consider
the
F
-prime-strip
∗
F
in
the
final
display
of
Definition
2.4,
(iii).
Next,
ob-
serve
that
the
“Picard
group”
constructions
“Pic
Φ
(−)”
and
“Pic
C
(−)”
of
[FrdI],
Theorem
5.1,
(i),
applied
to
any
object
of
the
global
realified
Frobenioid
∗
C
yield
canonically
isomorphic
groups
for
any
object
of
∗
C
.
In
particular,
it
makes
sense
to
speak
of
“Pic(
∗
C
)”.
Moreover,
it
follows
from
[FrdI],
Theorem
6.4,
(i),
(ii),
that
Pic(
∗
C
)
is
equipped
with
a
canonical
structure
of
ordered
monoid,
with
respect
to
which
it
is
isomorphic
to
R
[endowed
with
the
usual
additive
and
order
structures].
Relative
to
this
structure
of
ordered
monoid,
the
pilot
object
discussed
in
(i)
[cf.
also
the
discussion
of
[IUTchII],
Definition
4.9,
(viii)]
determines
a
negative
element
η
∗
C
∈
Pic(
∗
C
).
Thus,
one
verifies
immediately,
by
recalling
the
various
defini-
∼
tions
involved,
that
the
collection
of
data
“(M,
{M
v
}
v∈V
,
{ρ
M
v
:
M
v
→
M
}
v∈V
)”
constructed
in
(iii)
from
“M
”
is
already
sufficient
to
reconstruct,
i.e.,
by
taking
def
M
=
Pic(
∗
C
),
the
collection
of
data
∼
(
∗
C
,
Prime(
∗
C
)
→
V)
[cf.
the
notation
of
the
final
display
of
Definition
2.4,
(iii)],
while
the
collection
of
data
“(M,
{M
v
}
v∈V
,
{ρ
M
v
:
M
v
→
M
}
v∈V
)”
constructed
in
(iv)
from
the
def
pair
“(M,
η
M
)”
is
sufficient
to
reconstruct,
i.e.,
by
taking
M
=
Pic(
∗
C
)
and
def
η
M
=
η
∗
C
,
the
collection
of
data
∼
(
∗
C
,
Prime(
∗
C
)
→
V,
{Φ
∗
F
v
}
v∈V
,
{
∗
ρ
v
}
v∈V
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
91
where,
for
v
∈
V,
we
write
Φ
∗
F
v
for
the
[constant!]
divisor
monoid
[i.e.,
in
effect,
a
single
monoid
isomorphic
to
N
or
R
≥0
]
determined
by
the
Frobenioid
structure
[cf.
[FrdI],
Corollary
4.11,
(iii);
[FrdII],
Theorem
1.2,
(i)]
on
∗
F
v
[cf.
the
notation
of
the
final
display
of
Definition
2.4,
(i)].
(vi)
One
immediate
consequence
of
the
discussion
of
(v)
is
the
following:
If
one
starts
from
M
=
Pic(
∗
C
),
then
the
resulting
collection
of
data
∼
(
∗
C
,
Prime(
∗
C
)
→
V)
yields
a
common
container,
namely,
the
Frobenioid
∗
C
[regarded
as
an
object
reconstructed
from
M
=
Pic(
∗
C
)!],
in
which
distinct
choices
of
the
[negative!]
pilot
element
∈
M
=
Pic(
∗
C
)
—
hence
also
the
data
∼
(
∗
C
,
Prime(
∗
C
)
→
V,
{Φ
∗
F
v
}
v∈V
,
{
∗
ρ
v
}
v∈V
)
[which
may
be
thought
of
as
a
sort
of
“further
rigidification”
on
∗
C
]
recon-
structed
from
such
distinct
choices
of
pilot
element
—
may
be
compared
with
one
another.
By
contrast,
if
one
attempts
to
compare
the
constructions
of
(v)
applied
to
posi-
tive
and
negative
“η
M
∈
M
”
[i.e.,
which
amounts
to
reversing
the
order
structure
on
M
!],
then
already
the
corresponding
Frobenioids
“
∗
C
”
[i.e.,
attached
to
the
same
group
“Pic(
∗
C
)”,
but
with
reversed
order
structures!]
involve
pre-steps
[i.e.,
in
effect,
the
category-theoretic
version
of
the
notion
of
an
inclusion
of
line
bundles
—
cf.
[FrdI],
Definition
1.2,
(iii)]
going
in
opposite
directions.
That
is
to
say,
such
Frobenioids
may
only
be
compared
with
one
another
if
they
are
embedded
in
some
sort
of
larger
ambient
category
in
which
the
pre-steps
are
rendered
invertible;
but
this
already
implies
that
all
objects
arising
from
such
Frobenioids
become
iso-
morphic
in
the
ambient
category.
That
is
to
say,
working
in
such
a
larger
ambient
category
already
renders
any
sort
of
argument
that
requires
one
to
distinguish
dis-
tinct
elements
of
Pic(
∗
C
)
—
i.e.,
distinct
arithmetic
degrees/heights
of
arithmetic
line
bundles
—
meaningless
[cf.
the
discussion
of
positivity
in
Remark
2.1.1,
(v)].
92
SHINICHI
MOCHIZUKI
Section
3:
Multiradial
Logarithmic
Gaussian
Procession
Monoids
In
the
present
§3,
we
apply
the
theory
developed
thus
far
in
the
present
series
of
papers
to
give
[cf.
Theorem
3.11
below]
multiradial
algorithms
for
a
slightly
modified
version
of
the
Gaussian
monoids
discussed
in
[IUTchII],
§4.
This
modi-
fication
revolves
around
the
combinatorics
of
processions,
as
developed
in
[IUTchI],
§4,
§5,
§6,
and
is
necessary
in
order
to
establish
the
desired
multiradiality.
At
a
more
concrete
level,
these
combinatorics
require
one
to
apply
the
theory
of
tensor
packets
[cf.
Propositions
3.1,
3.2,
3.3,
3.4,
3.7,
3.9,
below].
Finally,
we
observe
in
Corollary
3.12
that
these
multiradial
algorithms
give
rise
to
certain
estimates
concerning
the
log-volumes
of
the
logarithmic
Gaussian
procession
monoids
that
occur.
This
observation
forms
the
starting
point
of
the
theory
to
be
developed
in
[IUTchIV].
In
the
following
discussion,
we
assume
that
we
have
been
given
initial
Θ-data
as
in
[IUTchI],
Definition
3.1.
Also,
we
shall
write
def
V
Q
=
V(Q)
[cf.
[IUTchI],
§0]
and
apply
the
notation
of
Definition
1.1
of
the
present
paper.
We
begin
by
discussing
the
theory
of
tensor
packets,
which
may
be
thought
of
as
a
sort
of
amalgamation
of
the
theory
of
log-shells
developed
in
§1
with
the
theory
of
processions
developed
in
[IUTchI],
§4,
§5,
§6.
Proposition
3.1.
(Local
Holomorphic
Tensor
Packets)
Let
{
α
F}
α∈A
=
{
α
F
v
}
v∈V
α∈A
be
an
n-capsule,
with
index
set
A,
of
F-prime-strips
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
§0;
[IUTchI],
Definition
5.2,
(i)].
Then
[cf.
the
notation
of
Definition
1.1,
(iii)]
for
V
v
|
v
Q
,
by
considering
invariants
with
respect
to
the
natural
action
of
various
open
subgroups
of
the
topological
group
α
Π
v
,
one
may
regard
log(
α
F
v
)
as
an
inductive
limit
of
topological
modules,
each
of
which
is
of
finite
dimension
over
Q
v
Q
;
we
shall
refer
to
the
correspondence
def
V
Q
v
Q
→
log(
α
F
v
Q
)
=
log(
α
F
v
)
V
v
|
v
Q
as
the
[1-]tensor
packet
associated
to
the
F-prime-strip
α
F
and
to
the
correspon-
dence
def
A
α
α
log(
F
v
Q
)
=
log(
F
v
α
)
V
Q
v
Q
→
log(
F
v
Q
)
=
α∈A
{v
α
}
α∈A
α∈A
—
where
the
tensor
products
are
to
be
understood
as
tensor
products
of
ind-topological
modules
[i.e.,
as
discussed
above],
and
the
direct
sum
is
over
all
collections
{v
α
}
α∈A
of
[not
necessarily
distinct!]
elements
v
α
∈
V
lying
over
v
Q
and
indexed
by
α
∈
A
—
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
93
as
the
[n-]tensor
packet
associated
to
the
collection
of
F-prime-strips
{
α
F}
α∈A
.
Then:
(i)
(Ring
Structures)
The
ind-topological
field
structures
on
the
var-
ious
log(
α
F
v
)
[cf.
Definition
1.1,
(i),
(ii),
(iii)],
for
α
∈
A,
determine
an
ind-
topological
ring
structure
on
log(
A
F
v
Q
)
with
respect
to
which
log(
A
F
v
Q
)
may
be
regarded
as
an
inductive
limit
of
direct
sums
of
ind-topological
fields.
Such
decompositions
as
direct
sums
of
ind-topological
fields
are
uniquely
determined
by
the
ind-topological
ring
structure
on
log(
A
F
v
Q
)
and,
moreover,
are
compatible,
for
α
∈
A,
with
the
natural
action
of
the
topological
group
α
Π
v
[where
V
v
|
v
Q
]
on
the
direct
summand
with
subscript
v
of
the
factor
labeled
α.
(ii)
(Integral
Structures)
Fix
elements
α
∈
A,
v
∈
V,
v
Q
∈
V
Q
such
that
v
|
v
Q
.
Relative
to
the
tensor
product
in
the
above
definition
of
log(
A
F
v
Q
),
write
def
log(
β
F
v
Q
)
⊆
log(
A
F
v
Q
)
log(
A,α
F
v
)
=
log(
α
F
v
)
⊗
β∈A\{α}
for
the
ind-topological
submodule
determined
by
the
tensor
product
of
the
factors
labeled
by
β
∈
A
\
{α}
with
the
tensor
product
of
the
direct
summand
with
subscript
v
of
the
factor
labeled
α.
Then
log(
A,α
F
v
)
forms
a
direct
summand
of
the
ind-
topological
ring
log(
A
F
v
Q
);
log(
A,α
F
v
)
may
be
regarded
as
an
inductive
limit
of
direct
sums
of
ind-topological
fields;
such
decompositions
as
direct
sums
of
ind-topological
fields
are
uniquely
determined
by
the
ind-topological
ring
structure
on
log(
A,α
F
v
).
Moreover,
by
forming
the
tensor
product
with
“1’s”
in
the
factors
labeled
by
β
∈
A
\
{α},
one
obtains
a
natural
injective
homomorphism
of
ind-topological
rings
log(
α
F
v
)
→
log(
A,α
F
v
)
that,
for
suitable
choices
[which
are,
in
fact,
cofinal]
of
objects
appearing
in
the
inductive
limit
descriptions
given
above
for
the
domain
and
codomain,
induces
an
isomorphism
of
such
an
object
in
the
domain
onto
each
of
the
direct
summand
ind-
topological
fields
of
the
object
in
the
codomain.
In
particular,
the
integral
structure
def
{0}
⊆
log(
α
F
v
)
Ψ
log(
α
F
v
)
=
Ψ
log(
α
F
v
)
[cf.
the
notation
of
Definition
1.1,
(i),
(ii)]
determines
integral
structures
on
each
of
the
direct
summand
ind-topological
fields
that
appear
in
the
inductive
limit
descriptions
of
log(
A,α
F
v
),
log(
A
F
v
Q
).
Proof.
The
various
assertions
of
Proposition
3.1
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions
[cf.
also
Remark
3.1.1,
(i),
below].
Remark
3.1.1.
def
(i)
Let
v
∈
V.
In
the
notation
of
[IUTchI],
Definition
3.1,
write
k
=
K
v
;
let
k
be
an
algebraic
closure
of
k.
Then,
roughly
speaking,
in
the
notation
of
Proposition
3.1,
∼
∼
log(
α
F
v
)
→
k;
Ψ
log(
α
F
v
)
→
O
k
;
94
SHINICHI
MOCHIZUKI
log(
A,α
F
v
)
∼
→
k
∼
→
lim
−→
k
⊇
lim
−→
O
k
—
i.e.,
one
verifies
immediately
that
each
ind-topological
field
log(
α
F
v
)
is
isomor-
phic
to
k;
each
log(
A,α
F
v
)
is
a
topological
tensor
product
[say,
over
Q]
of
copies
of
k,
hence
may
be
described
as
an
inductive
limit
of
direct
sums
of
copies
of
k;
each
Ψ
log(
α
F
v
)
is
a
copy
of
the
set
[i.e.,
a
ring,
when
v
∈
V
non
]
of
integers
O
k
⊆
k.
In
particular,
the
“integral
structures”
discussed
in
the
final
portion
of
Proposition
3.1,
(ii),
correspond
to
copies
of
O
k
contained
in
copies
of
k.
(ii)
Ultimately,
for
v
∈
V,
we
shall
be
interested
[cf.
Proposition
3.9,
(i),
(ii),
below]
in
considering
log-volumes
on
the
portion
of
log(
α
F
v
)
corresponding
to
K
v
.
On
the
other
hand,
let
us
recall
that
we
do
not
wish
to
consider
all
of
the
valuations
in
V(K).
That
is
to
say,
we
wish
to
restrict
ourselves
to
considering
the
∼
subset
V
⊆
V(K),
equipped
with
the
natural
bijection
V
→
V
mod
[cf.
[IUTchI],
Definition
3.1,
(e)],
which
we
wish
to
think
of
as
a
sort
of
“local
analytic
section”
[cf.
the
discussion
of
[IUTchI],
Remark
4.3.1,
(i)]
of
the
natural
morphism
Spec(K)
→
Spec(F
)
[or,
perhaps
more
precisely,
Spec(K)
→
Spec(F
mod
)].
In
particular,
it
will
be
necessary
to
consider
these
log-volumes
on
the
portion
of
log(
α
F
v
)
corresponding
to
K
v
relative
to
the
weight
[K
v
:
(F
mod
)
v
]
−1
,
where
we
write
v
∈
V
mod
for
the
element
determined
[via
the
natural
bijection
just
discussed]
by
v
[cf.
the
discussion
of
[IUTchI],
Example
3.5,
(i),
(ii),
(iii),
where
similar
factors
appear].
When,
moreover,
we
consider
direct
sums
over
all
v
∈
V
lying
over
a
given
v
Q
∈
V
Q
as
in
the
case
of
log(
α
F
v
Q
),
it
will
be
convenient
to
use
the
normalized
weight
1
[K
v
:
(F
mod
)
v
]
·
[(F
mod
)
w
:
Q
v
Q
]
V
mod
w|v
Q
—
i.e.,
normalized
so
that
multiplication
by
p
v
Q
affects
log-volumes
by
addition
or
non
subtraction
[that
is
to
say,
depending
on
whether
v
Q
∈
V
arc
Q
or
v
Q
∈
V
Q
]
of
the
quantity
log(p
v
Q
)
∈
R.
In
a
similar
vein,
when
we
consider
log-volumes
on
the
portion
of
log(
A
F
v
Q
)
corresponding
to
the
tensor
product
of
various
K
v
α
,
where
V
v
α
|
v
Q
,
it
will
be
necessary
to
consider
these
log-volumes
relative
to
the
weight
1
[K
v
α
:
(F
mod
)
v
α
]
α∈A
—
where
we
write
v
α
∈
V
mod
for
the
element
determined
by
v
α
.
When,
moreover,
we
consider
direct
sums
over
all
possible
choices
for
the
data
{v
α
}
α∈A
,
it
will
be
convenient
to
use
the
normalized
weight
α∈A
[K
v
α
:
(F
mod
)
v
α
]
·
1
{w
α
}
α∈A
α∈A
[(F
mod
)
w
α
:
Q
v
Q
]
—
where
the
sum
is
over
all
collections
{w
α
}
α∈A
of
[not
necessarily
distinct!]
el-
ements
w
α
∈
V
mod
lying
over
v
Q
and
indexed
by
α
∈
A.
Again,
these
normalized
weights
are
normalized
so
that
multiplication
by
p
v
Q
affects
log-volumes
by
addition
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
95
non
or
subtraction
[that
is
to
say,
depending
on
whether
v
Q
∈
V
arc
Q
or
v
Q
∈
V
Q
]
of
the
quantity
log(p
v
Q
)
∈
R.
(iii)
In
the
discussion
to
follow,
we
shall,
for
simplicity,
use
the
term
“measure
space”
to
refer
to
a
locally
compact
Hausdorff
topological
space
whose
topology
admits
a
countable
basis,
and
which
is
equipped
with
a
complete
Borel
measure
in
the
sense
of
[Royden],
Chapter
11,
§1;
[Royden],
Chapter
14,
§1.
In
particular,
one
may
speak
of
the
product
measure
space
[cf.
[Royden],
Chapter
12,
§4]
of
any
finite
nonempty
collection
of
measure
spaces.
Then
observe
that
care
must
be
exercised
when
considering
the
various
weighted
sums
of
log-volumes
discussed
in
(ii),
since,
unlike,
for
instance,
the
log-volumes
discussed
in
[item
(a)
of]
[AbsTopIII],
Proposition
5.7,
(i),
(ii),
such
weighted
sums
of
log-volumes
do
not,
in
general,
arise
as
some
positive
real
multiple
of
the
[natural]
logarithm
of
a
“volume”
or
“mea-
sure”
in
the
usual
sense
of
measure
theory.
In
particular,
when
considering
direct
sums
of
the
sort
that
appear
in
the
second
or
third
displays
of
the
statement
of
Proposition
3.1,
although
it
is
clear
from
the
definitions
how
to
compute
a
weighted
sum
of
log-volumes
of
the
sort
discussed
in
(ii)
in
the
case
of
a
region
that
arises
as
a
direct
product
of,
say,
compact
subsets
of
positive
measure
in
each
of
the
direct
summands
[i.e.,
since
the
volume/measure
of
such
a
compact
subset
may
be
computed
as
the
infimum
of
the
volume/measure
of
the
compact
open
subsets
that
contain
it],
it
is
not
immediately
clear
from
the
definitions
how
to
compute
such
a
weighted
sum
of
log-volumes
in
the
case
of
more
general
regions.
In
the
following,
for
ease
of
reference,
let
us
refer
to
such
a
region
that
arises
as
a
direct
product
of
compact
subsets
of
positive
measure
in
each
of
the
direct
summands
as
a
direct
product
region
and
to
a
region
that
arises
as
a
direct
product
of
relatively
compact
subsets
in
each
of
the
direct
summands
as
a
direct
product
pre-region.
Then
we
observe
in
the
remainder
of
the
present
Remark
3.1.1
that
although,
in
the
present
series
of
papers,
the
regions
that
will
actually
be
of
interest
in
the
development
of
the
theory
are,
in
fact,
direct
product
[pre-]regions,
in
which
case
the
com-
putation
of
weighted
sums
of
log-volumes
is
completely
straightforward
[cf.
also
the
discussion
of
Remark
3.9.7,
(ii),
(iii),
below],
in
fact,
weighted
sums
of
log-volumes
of
the
sort
discussed
in
(ii)
may
be
computed
for,
say,
arbitrary
Borel
sets
by
applying
the
elementary
construction
discussed
in
(iv)
below.
Here,
in
the
context
of
the
situation
discussed
in
the
final
portion
of
(ii),
we
note
that
this
construction
in
(iv)
below
is
applied
relative
to
the
following
given
data:
·
the
finite
set
“V
”
is
taken
to
be
the
direct
product
∼
(V
mod
)
v
Q
)
V
v
Q
(
→
α∈A
α∈A
96
SHINICHI
MOCHIZUKI
[where
the
subscript
“v
Q
”
denotes
the
fiber
over
v
Q
∈
V
Q
];
·
for
“v
∈
V
”,
the
cardinality
“N
v
”
is
taken
to
be
the
product
that
appears
in
the
discussion
of
(ii)
[K
v
α
:
(F
mod
)
v
α
]
α∈A
[where
we
think
of
“v
∈
V
”
as
a
collection
{v
α
}
α∈A
of
elements
of
V
v
Q
that
lies
over
a
collection
{v
α
}
α∈A
of
elements
of
(V
mod
)
v
Q
],
while
“M
v
”
is
taken
to
be
the
[radial,
if
v
Q
∈
V
Q
arc
]
portion
of
the
direct
summand
in
the
third
display
of
the
statement
of
Proposition
3.1
indexed
by
v
∈
V
that
corresponds
to
the
tensor
product
of
the
{K
v
α
}
α∈A
.
[By
the
“radial”
portion
of
a
topological
tensor
product
of
a
finite
collection
of
complex
archimedean
fields,
we
mean
the
direct
product
of
the
copies
of
R
>0
that
arise
by
forming
the
quotients
by
the
units
[i.e.,
copies
of
S
1
]
of
each
of
the
complex
archimedean
fields
that
appears
in
the
direct
sum
of
fields
[cf.
(i)]
that
arises
from
such
a
topological
tensor
product.]
Then
one
verifies
immediately
that,
in
the
case
of
“direct
product
regions”
[as
discussed
above],
the
result
of
multiplying
the
[natural]
logarithm
of
the
“E-weighted
measure
μ
E
(−)”
of
(iv)
by
a
suitable
normalization
factor
[i.e.,
a
suitable
positive
real
number]
yields
the
weighted
sums
of
log-volumes
discussed
in
(ii).
def
(iv)
Let
V
a
nonempty
finite
set;
E
=
{E
v
}
v∈V
a
collection
of
nonempty
def
finite
sets;
M
=
{(M
v
,
μ
v
)}
v∈V
a
collection
of
nonempty
measure
spaces
[cf.
the
discussion
of
(iii)
above].
For
v
∈
V
,
write
def
E
=
E
v
;
v
∈V
def
E
×
V
W
=
def
E
=
v
=
E
v
;
V
v
=
v
E
=
v
×
{v
}
V
v
∈V
—
where
the
first
arrow
“”
is
defined
by
the
condition
that,
for
v
∈
V
,
it
restricts
to
the
natural
projection
E
×{v
}
E
=
v
×{v
}
on
E
×{v
};
the
second
arrow
“”
is
defined
by
the
condition
that,
for
v
∈
V
,
it
restricts
to
the
natural
projection
E
=
v
×
{v
}
{v
}
on
E
=
v
×
{v
}.
If
W
w
→
v
∈
V
via
the
natural
surjection
def
W
V
just
discussed,
then
write
(M
w
,
μ
w
)
=
(M
v
,
μ
v
).
If
Z
is
a
subset
of
W
or
V
,
then
we
shall
write
def
M
Z
=
M
z
;
z∈Z
def
(M
E×V
⊇)
M
E∗V
=
def
M
E×V
=
(e,v)∈E×V
M
v
=
e∈E
{m
e,v
}
(e,v)∈E×V
|
m
e
,v
=
m
e
,v
,
∀(e
,
e
)
∈
E
×
E
=
v
E
⊆
E
×
E
∼
M
V
;
∼
→
M
W
—
where
the
bijection
M
E∗V
→
M
W
is
the
map
induced
by
the
various
natural
projections
E
E
=
v
that
constitute
the
natural
projection
E
×
V
W
;
this
∼
bijection
M
E∗V
→
M
W
is
easily
verified
to
be
a
homeomorphism.
Thus,
M
W
,
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
97
M
V
,
and
M
E×V
are
equipped
with
natural
product
measure
space
structures;
the
∼
bijection
M
E∗V
→
M
W
,
together
with
the
measure
space
structure
on
M
W
,
induces
a
measure
space
structure
on
M
E∗V
.
In
particular,
if
S
⊆
M
V
is
any
Borel
set,
then
the
product
S
⊆
M
E×V
e∈E
is
a
Borel
set
of
M
E×V
;
the
intersection
of
this
product
with
M
E∗V
def
S
E
=
S
M
E∗V
⊆
M
E∗V
e∈E
∼
is
a
Borel
set
of
M
E∗V
(
→
M
W
).
Thus,
in
summary,
for
any
Borel
set
S
⊆
M
V
,
one
may
speak
of
the
“E-weighted
measure”
{+∞}
μ
E
(S)
∈
R
≥0
∼
of
S,
i.e.,
the
measure,
relative
to
the
measure
space
structure
of
M
E∗V
(
→
M
W
),
of
S
E
.
Since,
moreover,
one
verifies
immediately
that
the
above
construction
is
functorial
with
respect
to
isomorphisms
of
the
given
data
(V,
E,
M),
it
follows
def
immediately
that,
in
fact,
μ
E
(−)
is
completely
determined
by
the
cardinalities
N
=
{N
v
}
v∈V
of
the
finite
sets
E
=
{E
v
}
v∈V
,
i.e.,
by
the
data
(V,
N
,
M).
Finally,
we
observe
that
when
S
⊆
M
V
is
a
“direct
product
region”
[cf.
the
discussion
of
(iii)],
i.e.,
a
set
of
the
form
v∈V
S
v
,
where
S
v
⊆
M
v
is
a
compact
subset
of
positive
measure,
then
a
straightforward
computation
reveals
that
log
log
1
1
·
μ
(S)
=
E
N
E
N
v
·
μ
v
(S
v
)
v∈V
—
where
we
write
N
E
=
v∈V
N
v
,
and
each
superscript
“log”
denotes
the
natural
logarithm
of
the
corresponding
quantity
without
a
superscript.
Remark
3.1.2.
The
constructions
involving
local
holomorphic
tensor
packets
given
in
Proposition
3.1
may
be
applied
to
the
capsules
that
appear
in
the
various
F-prime-strip
processions
obtained
by
considering
the
evident
F-prime-strip
analogues
[cf.
[IUTchI],
Remark
5.6.1;
[IUTchI],
Remark
6.12.1]
of
the
holomor-
phic
processions
discussed
in
[IUTchI],
Proposition
4.11,
(i);
[IUTchI],
Proposi-
tion
6.9,
(i).
Proposition
3.2.
(Local
Mono-analytic
Tensor
Packets)
Let
α
α
{
D
}
α∈A
=
{
D
v
}
v∈V
α∈A
be
an
n-capsule,
with
index
set
A,
of
D
-prime-strips
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
§0;
[IUTchI],
Definition
4.1,
(iii)].
Then
[cf.
the
notation
of
Proposition
1.2,
(vi),
(vii)]
we
shall
refer
to
the
correspondence
V
Q
v
Q
→
log(
α
D
v
Q
)
=
def
V
v
|
v
Q
log(
α
D
v
)
98
SHINICHI
MOCHIZUKI
as
the
[1-]tensor
packet
associated
to
the
D
-prime-strip
α
D
and
to
the
corre-
spondence
def
log(
α
D
v
Q
)
V
Q
v
Q
→
log(
A
D
v
Q
)
=
α∈A
—
where
the
tensor
product
is
to
be
understood
as
a
tensor
product
of
ind-topological
modules
—
as
the
[n-]tensor
packet
associated
to
the
collection
of
D
-prime-strips
{
α
D
}
α∈A
.
For
α
∈
A,
v
∈
V,
v
Q
∈
V
Q
such
that
v
|
v
Q
,
we
shall
write
log(
A,α
D
v
)
⊆
log(
A
D
v
Q
)
for
the
ind-topological
submodule
determined
by
the
tensor
product
of
the
factors
labeled
by
β
∈
A
\
{α}
with
the
tensor
product
of
the
direct
summand
with
subscript
v
of
the
factor
labeled
α
[cf.
Proposition
3.1,
(ii)].
If
the
capsule
of
D
-prime-strips
{
α
D
}
α∈A
arises
from
a
capsule
of
F
×μ
-prime-strips
{
α
F
×μ
}
α∈A
=
{
α
F
v
×μ
}
v∈V
α∈A
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
§0;
[IUTchII],
Definition
4.9,
(vii)],
then
we
shall
use
similar
notation
to
the
notation
just
introduced
concerning
{
α
D
}
α∈A
to
denote
objects
associated
to
{
α
F
×μ
}
α∈A
,
i.e.,
by
replacing
“D
”
in
the
above
notational
conventions
by
“F
×μ
”
[cf.
also
the
notation
of
Proposition
1.2,
(vi),
(vii)].
Then:
(i)
(Mono-analytic/Holomorphic
Compatibility)
Suppose
that
the
cap-
sule
of
D
-prime-strips
{
α
D
}
α∈A
arises
from
the
capsule
of
F-prime-strips
{
α
F}
α∈A
of
Proposition
3.1;
write
{
α
F
×μ
}
α∈A
for
the
capsule
of
F
×μ
-prime-strips
associ-
∼
∼
ated
to
{
α
F}
α∈A
.
Then
the
poly-isomorphisms
“log(
†
D
v
)
→
log(
†
F
v
×μ
)
→
log(
†
F
v
)”
of
Proposition
1.2,
(vi),
(vii),
induce
natural
poly-isomorphisms
of
ind-topologi-
cal
modules
∼
∼
)
→
log(
α
F
v
Q
);
log(
α
D
v
Q
)
→
log(
α
F
v
×μ
Q
∼
∼
∼
log(
A
D
v
Q
)
→
log(
A
F
v
×μ
)
→
log(
A
F
v
Q
)
Q
∼
log(
A,α
D
v
)
→
log(
A,α
F
v
×μ
)
→
log(
A,α
F
v
)
between
the
various
“mono-analytic”
tensor
packets
of
the
present
Proposition
3.2
and
the
“holomorphic”
tensor
packets
of
Proposition
3.1.
(ii)
(Integral
Structures)
If
V
v
|
v
Q
∈
V
non
Q
,
then
the
mono-analytic
log-shells
“I
†
D
v
”
of
Proposition
1.2,
(vi),
determine
[i.e.,
by
forming
suitable
direct
sums
and
tensor
products]
topological
submodules
I(
α
D
v
Q
)
⊆
log(
α
D
v
Q
);
I(
A
D
v
Q
)
⊆
log(
A
D
v
Q
);
I(
A,α
D
v
)
⊆
log(
A,α
D
v
)
—
which
may
be
regarded
as
integral
structures
on
the
Q-spans
of
these
sub-
modules.
If
V
v
|
v
Q
∈
V
arc
Q
,
then
by
regarding
the
mono-analytic
log-shell
“I
†
D
v
”
of
Proposition
1.2,
(vii),
as
the
“closed
unit
ball”
of
a
Hermitian
metric
on
“log(
†
D
v
)”,
and
considering
the
induced
direct
sum
Hermitian
metric
on
log(
α
D
v
Q
),
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
99
together
with
the
induced
tensor
product
Hermitian
metric
on
log(
A
D
v
Q
),
one
ob-
tains
Hermitian
metrics
on
log(
α
D
v
Q
),
log(
A
D
v
Q
),
and
log(
A,α
D
v
),
whose
associ-
ated
closed
unit
balls
I(
α
D
v
Q
)
⊆
log(
α
D
v
Q
);
I(
A
D
v
Q
)
⊆
log(
A
D
v
Q
);
I(
A,α
D
v
)
⊆
log(
A,α
D
v
)
may
be
regarded
as
integral
structures
on
log(
α
D
v
Q
),
log(
A
D
v
Q
),
and
log(
A,α
D
v
),
respectively.
For
arbitrary
V
v
|
v
Q
∈
V
Q
,
we
shall
denote
by
“I
Q
((−))”
the
Q-
span
of
“I((−))”;
also,
we
shall
apply
this
notation
involving
“I((−))”,
“I
Q
((−))”
with
“D
”
replaced
by
“F”
or
“F
×μ
”
for
the
various
objects
obtained
from
the
“D
-versions”
discussed
above
by
applying
the
natural
poly-isomorphisms
of
(i).
Proof.
The
various
assertions
of
Proposition
3.2
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
3.2.1.
The
issue
of
estimating
the
discrepancy
between
the
holo-
morphic
integral
structures
of
Proposition
3.1,
(ii),
and
the
mono-analytic
in-
tegral
structures
of
Proposition
3.2,
(ii),
will
form
one
of
the
main
topics
to
be
discussed
in
[IUTchIV]
—
cf.
also
Remark
3.9.1
below.
Remark
3.2.2.
The
constructions
involving
local
mono-analytic
tensor
pack-
ets
given
in
Proposition
3.2
may
be
applied
to
the
capsules
that
appear
in
the
various
D
-prime-strip
processions
—
i.e.,
mono-analytic
processions
—
discussed
in
[IUTchI],
Proposition
4.11,
(ii);
[IUTchI],
Proposition
6.9,
(ii).
Proposition
3.3.
(Global
Tensor
Packets)
Let
†
HT
Θ
±ell
NF
be
a
Θ
±ell
NF-Hodge
theater
[relative
to
the
given
initial
Θ-data]
—
cf.
[IUTchI],
±ell
Definition
6.13,
(i).
Thus,
†
HT
Θ
NF
determines
ΘNF-
and
Θ
±ell
-Hodge
theaters
±ell
†
HT
ΘNF
,
†
HT
Θ
as
in
[IUTchII],
Corollary
4.8.
Let
{
α
F}
α∈A
be
an
n-capsule
of
F-prime-strips
as
in
Proposition
3.1.
Suppose,
further,
that
A
is
a
subset
of
the
index
set
J
that
appears
in
the
ΘNF-Hodge
theater
†
HT
ΘNF
,
and
that,
for
each
α
∈
A,
we
are
given
a
log-link
α
F
log
−→
†
F
α
∼
—
i.e.,
a
poly-isomorphism
of
F-prime-strips
log(
α
F)
→
†
F
α
[cf.
Definition
1.1,
(iii)].
Next,
recall
the
field
†
M
mod
discussed
in
[IUTchII],
Corollary
4.8,
(i);
thus,
one
also
has,
for
j
∈
J,
a
labeled
version
(
†
M
mod
)
j
of
this
field
[cf.
[IUTchII],
Corollary
4.8,
(ii)].
We
shall
refer
to
(
†
M
mod
)
A
=
def
α∈A
(
†
M
mod
)
α
100
SHINICHI
MOCHIZUKI
—
where
the
tensor
product
is
to
be
understood
as
a
tensor
product
of
modules
—
as
the
global
[n-]tensor
packet
associated
to
the
subset
A
⊆
J
and
the
Θ
±ell
NF-
±ell
Hodge
theater
†
HT
Θ
NF
.
(i)
(Ring
Structures)
The
field
structure
on
the
various
(
†
M
mod
)
α
,
for
α
∈
A,
determine
a
ring
structure
on
(
†
M
mod
)
A
with
respect
to
which
(
†
M
mod
)
A
decomposes,
uniquely,
as
a
direct
sum
of
number
fields.
Moreover,
the
various
)
j
→
†
F
j
”
considered
in
[IUTchII],
Corollary
localization
functors
“(
†
F
mod
4.8,
(iii),
determine,
by
composing
with
the
given
log-links,
a
natural
injective
localization
ring
homomorphism
(
†
M
mod
)
A
→
def
log(
A
F
V
Q
)
=
log(
A
F
v
Q
)
v
Q
∈V
Q
to
the
product
of
the
local
holomorphic
tensor
packets
considered
in
Proposition
3.1.
(ii)
(Integral
Structures)
Fix
an
element
α
∈
A.
Then
by
forming
the
tensor
product
with
“1’s”
in
the
factors
labeled
by
β
∈
A
\
{α},
one
obtains
a
natural
ring
homomorphism
(
†
M
mod
)
α
→
(
†
M
mod
)
A
that
induces
an
isomorphism
of
the
domain
onto
a
subfield
of
each
of
the
direct
summand
number
fields
of
the
codomain.
For
each
v
Q
∈
V
Q
,
this
homomorphism
is
compatible,
in
the
evident
sense,
relative
to
the
localization
homomorphism
of
(i),
with
the
natural
homomorphism
of
ind-topological
rings
considered
in
Propo-
sition
3.1,
(ii).
Moreover,
for
each
v
Q
∈
V
non
Q
,
the
composite
of
the
above
dis-
played
homomorphism
with
the
component
at
v
Q
of
the
localization
homomorphism
of
(i)
maps
the
ring
of
integers
of
the
number
field
(
†
M
mod
)
α
into
the
submod-
ule
constituted
by
the
integral
structure
on
log(
A
F
v
Q
)
considered
in
Proposition
3.1,
(ii);
for
each
v
Q
∈
V
arc
Q
,
the
composite
of
the
above
displayed
homomorphism
with
the
component
at
v
Q
of
the
localization
homomorphism
of
(i)
maps
the
set
of
archimedean
integers
[i.e.,
elements
of
absolute
value
≤
1
at
all
archimedean
primes]
of
the
number
field
(
†
M
mod
)
α
into
the
direct
product
of
subsets
constituted
by
the
integral
structures
considered
in
Proposition
3.1,
(ii),
on
the
various
direct
summand
ind-topological
fields
of
log(
A
F
v
Q
).
Proof.
The
various
assertions
of
Proposition
3.3
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
3.3.1.
One
may
perform
analogous
constructions
to
the
constructions
of
Proposition
3.3
for
the
fields
“M
mod
(
†
D
)
j
”
of
[IUTchII],
Corollary
4.7,
(ii)
[cf.
also
the
localization
functors
of
[IUTchII],
Corollary
4.7,
(iii)],
constructed
from
±ell
the
associated
D-Θ
±ell
NF-Hodge
theater
†
HT
D-Θ
NF
.
These
constructions
are
compatible
with
the
corresponding
constructions
of
Proposition
3.3,
in
the
evident
sense,
relative
to
the
various
labeled
Kummer-theoretic
isomorphisms
of
[IUTchII],
Corollary
4.8,
(ii).
We
leave
the
routine
details
to
the
reader.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
101
Remark
3.3.2.
(i)
One
may
consider
the
image
of
the
localization
homomorphism
of
Propo-
sition
3.3,
(i),
in
the
case
of
the
various
local
holomorphic
tensor
packets
arising
from
processions,
as
discussed
in
Remark
3.1.2.
Indeed,
at
the
level
of
the
labels
involved,
this
is
immediate
in
the
case
of
the
“F
l
-processions”
of
[IUTchI],
Propo-
sition
4.11,
(i).
On
the
other
hand,
in
the
case
of
the
“|F
l
|-processions”
of
[IUTchI],
Proposition
6.9,
(i),
this
may
be
achieved
by
applying
the
identifying
isomorphisms
between
the
zero
label
0
∈
|F
l
|
and
the
diagonal
label
F
l
associated
to
F
l
dis-
cussed
in
[the
final
display
of]
[IUTchII],
Corollary
4.6,
(iii)
[cf.
also
[IUTchII],
Corollary
4.8,
(ii)].
(ii)
In
a
similar
vein,
one
may
compose
the
“D-Θ
±ell
NF-Hodge
theater
version”
discussed
in
Remark
3.3.1
of
the
localization
homomorphism
of
Proposition
3.3,
(i),
with
the
product
over
v
Q
∈
V
Q
of
the
inverses
of
the
upper
right-hand
dis-
played
isomorphisms
at
v
Q
of
Proposition
3.2,
(i),
and
then
consider
the
image
of
this
composite
morphism
in
the
case
of
the
various
local
mono-analytic
tensor
packets
arising
from
processions,
as
discussed
in
Remark
3.2.2.
Just
as
in
the
holomorphic
case
discussed
in
(i),
in
the
case
of
the
“|F
l
|-processions”
of
[IUTchI],
Proposition
6.9,
(ii),
this
obliges
one
to
apply
the
identifying
isomorphisms
between
the
zero
label
0
∈
|F
l
|
and
the
diagonal
label
F
l
associated
to
F
l
discussed
in
[the
final
display
of]
[IUTchII],
Corollary
4.5,
(iii).
(iii)
The
various
images
of
global
tensor
packets
discussed
in
(i)
and
(ii)
above
may
be
identified
—
i.e.,
in
light
of
the
injectivity
of
the
homomorphisms
applied
to
construct
these
images
—
with
the
global
tensor
packets
themselves.
These
local
holomorphic/local
mono-analytic
global
tensor
packet
images
will
play
a
central
role
in
the
development
of
the
theory
of
the
present
§3
[cf.,
e.g.,
Proposition
3.7,
below].
Remark
3.3.3.
The
log-shifted
nature
of
the
localization
homomorphism
of
Proposition
3.3,
(i),
will
play
a
crucial
role
in
the
development
of
the
theory
of
present
§3
—
cf.
the
discussion
of
[IUTchII],
Remark
4.8.2,
(i),
(iii).
/
±
S
±
1
→
/
±
/
±
S
±
1+1=2
2
2
q
1
q
j
→
...
→
/
±
/
±
.
.
.
/
±
q
(l
)
→
...
→
S
±
j+1
/
±
/
±
.
.
.
.
.
.
/
±
S
±
1+l
=l
±
Fig.
3.1:
Splitting
monoids
of
LGP-monoids
acting
on
tensor
packets
Proposition
3.4.
(Local
Packet-theoretic
Frobenioids)
(i)
(Single
Packet
Monoids)
In
the
situation
of
Proposition
3.1,
fix
elements
α
∈
A,
v
∈
V,
v
Q
∈
V
Q
such
that
v
|
v
Q
.
Then
the
operation
of
forming
the
image
via
the
natural
homomorphism
log(
α
F
v
)
→
log(
A,α
F
v
)
[cf.
Proposition
3.1,
(ii)]
102
SHINICHI
MOCHIZUKI
of
the
monoid
Ψ
log(
α
F
v
)
[cf.
the
notation
of
Definition
1.1,
(i),
(ii)],
together
with
R
its
submonoid
of
units
Ψ
×
log(
α
F
v
)
and
realification
Ψ
log(
α
F
v
)
,
determines
monoids
Ψ
×
,
log(
A,α
F
v
)
Ψ
log(
A,α
F
v
)
,
Ψ
R
log(
A,α
F
v
)
—
which
are
equipped
with
G
v
(
α
Π
v
)-actions
when
v
∈
V
non
and,
in
the
case
of
the
first
displayed
monoid,
with
a
pair
consisting
of
an
Aut-holomorphic
orbispace
and
a
Kummer
structure
when
v
∈
V
arc
.
We
shall
think
of
these
monoids
as
[possibly
realified]
subquotients
of
log(
A,α
F
v
)
that
act
[multiplicatively]
on
suitable
[possibly
realified]
subquotients
of
log(
A,α
F
v
).
In
particular,
when
v
∈
V
non
,
the
first
displayed
monoid,
together
with
its
α
Π
v
-
action,
determine
a
Frobenioid
equipped
with
a
natural
isomorphism
to
log(
α
F
v
);
when
v
∈
V
arc
,
the
first
displayed
monoid,
together
with
its
Aut-holomorphic
orbis-
pace
and
Kummer
structure,
determine
a
collection
of
data
equipped
with
a
natural
isomorphism
to
log(
α
F
v
).
(ii)
(Local
Logarithmic
Gaussian
Procession
Monoids)
Let
‡
HT
Θ
±ell
NF
log
−→
†
±ell
HT
Θ
NF
be
a
log-link
of
Θ
±ell
NF-Hodge
theaters
as
in
Proposition
1.3,
(i)
[cf.
also
the
situation
of
Proposition
3.3].
Consider
the
F-prime-strip
processions
that
arise
as
the
F-prime-strip
analogues
[cf.
Remark
3.1.2;
[IUTchI],
Remark
6.12.1]
of
the
holomorphic
processions
discussed
in
[IUTchI],
Proposition
6.9,
(i),
when
the
functor
of
[IUTchI],
Proposition
6.9,
(i),
is
applied
to
the
Θ
±
-bridges
associated
±ell
±ell
to
†
HT
Θ
NF
,
‡
HT
Θ
NF
;
we
shall
refer
to
such
processions
as
“†-”
or
“‡-”
processions.
Here,
we
recall
that
for
j
∈
{1,
.
.
.
,
l
},
the
index
set
of
the
(j
+
1)-
capsule
that
appears
in
such
a
procession
is
denoted
S
±
j+1
.
Then
by
applying
the
various
constructions
of
“single
packet
monoids”
given
in
(i)
in
the
case
of
the
various
capsules
of
F-prime-strips
that
appear
in
a
holomorphic
‡-procession
—
i.e.,
more
precisely,
in
the
case
of
the
label
j
∈
{1,
.
.
.
,
l
}
[which
we
shall
occasionally
identify
with
its
image
in
F
l
⊆
|F
l
|]
that
appears
in
the
(j
+1)-capsule
of
the
‡-procession
—
to
the
pull-backs,
via
the
poly-isomorphisms
that
appear
in
the
definition
[cf.
Definition
1.1,
(iii)]
of
the
given
log-link,
of
the
[collections
of
]
monoids
equipped
with
actions
by
topological
groups
when
v
∈
V
non
and
splittings
[up
to
torsion,
when
v
∈
V
bad
]
Ψ
F
gau
(
†
HT
Θ
)
v
,
∞
Ψ
F
gau
(
†
HT
Θ
)
v
of
[IUTchII],
Corollary
4.6,
(iv),
for
v
∈
V,
one
obtains
a
functorial
algorithm
±ell
±ell
log
in
the
log-link
of
Θ
±ell
NF-Hodge
theaters
‡
HT
Θ
NF
−→
†
HT
Θ
NF
for
constructing
[collections
of
]
monoids
equipped
with
actions
by
topological
groups
when
v
∈
V
non
and
splittings
[up
to
torsion,
when
v
∈
V
bad
]
V
v
→
Ψ
F
LGP
(
†
HT
Θ
±ell
NF
)
v
;
V
v
→
∞
Ψ
F
LGP
(
†
HT
Θ
±ell
NF
)
v
—
which
we
refer
to
as
“[local]
LGP-monoids”,
or
“logarithmic
Gaussian
proces-
±ell
sion
monoids”
[cf.
Fig.
3.1
above].
Here,
we
note
that
the
notation
“(
†
HT
Θ
NF
)”
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
103
constitutes
a
slight
abuse
of
notation.
Also,
we
note
that
this
functorial
algo-
rithm
requires
one
to
apply
the
compatibility
of
the
given
log-link
with
the
F
±
l
-
bad
symmetrizing
isomorphisms
involved
[cf.
Remark
1.3.2].
For
v
∈
V
,
the
component
labeled
j
∈
{1,
.
.
.
,
l
}
of
the
submonoid
of
Galois
invariants
[cf.
(i)]
±ell
of
the
entire
LGP-monoid
Ψ
F
LGP
(
†
HT
Θ
NF
)
v
is
a
subset
of
±
I
Q
(
S
j+1
,j;‡
F
v
)
[i.e.,
where
the
notation
“;
‡”
denotes
the
result
of
applying
the
discussion
of
(i)
to
the
case
of
F-prime-strips
labeled
“‡”;
cf.
also
the
notational
conventions
of
±
Proposition
3.2,
(ii)]
that
acts
multiplicatively
on
I
Q
(
S
j+1
,j;‡
F
v
)
[cf.
the
construc-
tions
of
[IUTchII],
Corollary
3.6,
(ii)].
For
any
v
∈
V,
the
component
labeled
j
∈
{1,
.
.
.
,
l
}
of
the
submodule
of
Galois
invariants
[cf.
(i)
when
v
∈
V
non
;
this
±ell
Galois
action
is
trivial
when
v
∈
V
arc
]
of
the
unit
portion
Ψ
F
LGP
(
†
HT
Θ
NF
)
×
v
of
such
an
LGP-monoid
is
a
subset
of
±
I
Q
(
S
j+1
,j;‡
F
v
)
[cf.
the
discussion
of
(i);
the
notational
conventions
of
Proposition
3.2,
(ii)]
that
±
acts
multiplicatively
on
I
Q
(
S
j+1
,j;‡
F
v
)
[cf.
the
constructions
of
[IUTchII],
Corollary
3.6,
(ii);
[IUTchII],
Proposition
4.2,
(iv);
[IUTchII],
Proposition
4.4,
(iv)].
Proof.
The
various
assertions
of
Proposition
3.4
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Proposition
3.5.
(Kummer
Theory
and
Upper
Semi-compatibility
for
±ell
Vertically
Coric
Local
LGP-Monoids)
Let
{
n,m
HT
Θ
NF
}
n,m∈Z
be
a
collec-
tion
of
distinct
Θ
±ell
NF-Hodge
theaters
[relative
to
the
given
initial
Θ-data]
—
which
we
think
of
as
arising
from
a
Gaussian
log-theta-lattice
[cf.
Definition
1.4].
For
each
n
∈
Z,
write
n,◦
±ell
HT
D-Θ
NF
for
the
D-Θ
±ell
NF-Hodge
theater
determined,
up
to
isomorphism,
by
the
various
±ell
n,m
HT
Θ
NF
,
where
m
∈
Z,
via
the
vertical
coricity
of
Theorem
1.5,
(i).
(i)
(Vertically
Coric
Local
LGP-Monoids
and
Associated
Kummer
Theory)
Write
F(
n,◦
D
)
t
for
the
F-prime-strip
associated
[cf.
[IUTchII],
Remark
4.5.1,
(i)]
to
the
labeled
collection
of
monoids
“Ψ
cns
(
n,◦
D
)
t
”
of
[IUTchII],
Corollary
4.5,
(iii)
[i.e.,
where
we
take
“†”
to
be
“n,
◦”].
Recall
the
constructions
of
Proposition
3.4,
(ii),
involving
F-prime-strip
processions.
Then
by
applying
these
constructions
to
the
F-prime-
strips
“F(
n,◦
D
)
t
”
and
the
various
full
log-links
associated
[cf.
the
discussion
of
Proposition
1.2,
(ix)]
to
these
F-prime-strips
—
which
we
consider
in
a
fashion
compatible
with
the
F
±
l
-symmetries
involved
[cf.
Remark
1.3.2;
Proposition
104
SHINICHI
MOCHIZUKI
3.4,
(ii)]
—
we
obtain
a
functorial
algorithm
in
the
D-Θ
±ell
NF-Hodge
theater
±ell
n,◦
HT
D-Θ
NF
for
constructing
[collections
of
]
monoids
±ell
V
v
→
Ψ
LGP
(
n,◦
HT
D-Θ
NF
V
v
→
∞
Ψ
LGP
(
n,◦
HT
D-Θ
)
v
;
±ell
NF
)
v
equipped
with
actions
by
topological
groups
when
v
∈
V
non
and
splittings
[up
to
torsion,
when
v
∈
V
bad
]
—
which
we
refer
to
as
“vertically
coric
[local]
LGP-
monoids”.
For
each
n,
m
∈
Z,
this
functorial
algorithm
is
compatible
[in
the
evident
sense]
with
the
functorial
algorithm
of
Proposition
3.4,
(ii)
—
i.e.,
where
we
take
“†”
to
be
“n,
m”
and
“‡”
to
be
“n,
m
−
1”
—
relative
to
the
Kummer
isomorphisms
of
labeled
data
Ψ
cns
(
n,m
F
)
t
∼
→
Ψ
cns
(
n,◦
D
)
t
of
[IUTchII],
Corollary
4.6,
(iii),
and
the
evident
identification,
for
m
=
m,
m
−
1,
of
n,m
F
t
[i.e.,
the
F-prime-strip
that
appears
in
the
associated
Θ
±
-bridge]
with
the
F-prime-strip
associated
to
Ψ
cns
(
n,m
F
)
t
.
In
particular,
for
each
n,
m
∈
Z,
we
obtain
Kummer
isomorphisms
of
[collections
of
]
monoids
Ψ
F
LGP
(
n,m
HT
Θ
∞
Ψ
F
LGP
(
n,m
HT
Θ
±ell
±ell
∼
±ell
NF
)
v
→
Ψ
LGP
(
n,◦
HT
D-Θ
NF
)
v
→
∼
∞
Ψ
LGP
(
n,◦
NF
±ell
HT
D-Θ
)
v
NF
)
v
equipped
with
actions
by
topological
groups
when
v
∈
V
non
and
splittings
[up
to
torsion,
when
v
∈
V
bad
],
for
v
∈
V.
(ii)
(Upper
Semi-compatibility)
The
Kummer
isomorphisms
of
the
final
two
displays
of
(i)
are
“upper
semi-compatible”
—
cf.
the
discussion
of
“up-
per
semi-commutativity”
in
Remark
1.2.2,
(iii)
—
with
the
various
log-links
of
±ell
log
±ell
Θ
±ell
NF-Hodge
theaters
n,m−1
HT
Θ
NF
−→
n,m
HT
Θ
NF
[where
m
∈
Z]
of
the
Gaussian
log-theta-lattice
under
consideration
in
the
following
sense.
Let
j
∈
{0,
1,
.
.
.
,
l
}.
Then:
(a)
(Nonarchimedean
Primes)
For
v
Q
∈
V
non
Q
,
the
topological
module
±
I(
S
j+1
F(
n,◦
D
)
v
Q
)
—
i.e.,
that
arises
from
applying
the
constructions
of
Proposition
3.4,
(ii)
[where
we
allow
“j”
to
be
0],
in
the
vertically
coric
context
of
(i)
above
[cf.
also
the
notational
conventions
of
Proposition
3.2,
(ii)]
—
contains
the
images
of
the
submodules
of
Galois
invariants
[where
we
recall
the
Galois
actions
that
appear
in
the
data
of
[IUTchII],
Corollary
4.6,
(i),
(iii)]
of
the
groups
of
units
(Ψ
cns
(
n,m
F
)
|t|
)
×
v
,
for
V
v
|
v
Q
and
|t|
∈
{0,
.
.
.
,
j},
via
both
(1)
the
tensor
product,
over
such
|t|,
of
the
[relevant]
Kummer
isomorphisms
of
(i),
and
(2)
the
tensor
product,
over
such
|t|,
of
the
pre-composite
of
these
Kummer
isomorphisms
with
the
m
-th
iterates
[cf.
Remark
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
1.1.1]
of
the
log-links,
for
m
≥
1,
of
the
n-th
column
of
the
Gaussian
log-theta-lattice
under
consideration
[cf.
the
discussion
of
Remark
1.2.2,
(i),
(iii)].
(b)
(Archimedean
Primes)
For
v
Q
∈
V
arc
Q
,
the
closed
unit
ball
±
I(
S
j+1
F(
n,◦
D
)
v
Q
)
—
i.e.,
that
arises
from
applying
the
constructions
of
Proposition
3.4,
(ii)
[where
we
allow
“j”
to
be
0],
in
the
vertically
coric
context
of
(i)
above
[cf.
also
the
notational
conventions
of
Proposition
3.2,
(ii)]
—
contains
the
image,
via
the
tensor
product,
over
|t|
∈
{0,
.
.
.
,
j},
of
the
[relevant]
Kummer
isomorphisms
of
(i),
of
both
(1)
the
groups
of
units
(Ψ
cns
(
n,m
F
)
|t|
)
×
v
,
for
V
v
|
v
Q
,
and
gp
(2)
the
closed
balls
of
radius
π
inside
(Ψ
cns
(
n,m
F
)
|t|
)
v
[cf.
the
notational
conventions
of
Definition
1.1],
for
V
v
|
v
Q
.
Here,
we
recall
from
the
discussion
of
Remark
1.2.2,
(ii),
(iii),
that,
if
we
regard
each
log-link
as
a
correspondence
that
only
concerns
the
units
that
appear
in
its
domain
[cf.
Remark
1.1.1],
then
a
closed
ball
as
in
(2)
contains,
for
each
m
≥
1,
a
subset
that
surjects,
via
the
m
-th
iterate
of
the
log-link
of
the
n-th
column
of
the
Gaussian
log-theta-lattice
under
consideration,
onto
the
subset
of
the
group
of
units
(Ψ
cns
(
n,m−m
F
)
|t|
)
×
v
on
which
this
iterate
is
defined.
(c)
(Bad
Primes)
Let
v
∈
V
bad
;
suppose
that
j
=
0.
Recall
that
the
various
monoids
“Ψ
F
LGP
(−)
v
”,
“
∞
Ψ
F
LGP
(−)
v
”
constructed
in
Proposition
3.4,
(ii),
as
well
as
the
monoids
“Ψ
LGP
(−)
v
”,
“
∞
Ψ
LGP
(−)
v
”
constructed
in
(i)
above,
are
equipped
with
natural
splittings
up
to
torsion.
Write
Ψ
⊥
F
LGP
(−)
v
⊆
Ψ
F
LGP
(−)
v
;
⊥
∞
Ψ
F
LGP
(−)
v
⊆
∞
Ψ
F
LGP
(−)
v
Ψ
⊥
LGP
(−)
v
⊆
Ψ
LGP
(−)
v
;
⊥
∞
Ψ
LGP
(−)
v
⊆
∞
Ψ
LGP
(−)
v
for
the
submonoids
corresponding
to
these
splittings
[cf.
the
submonoids
“O
⊥
(−)
⊆
O
(−)”
discussed
in
Definition
2.4,
(i),
in
the
case
of
“Ψ
⊥
”;
the
notational
conventions
of
Theorem
2.2,
(ii),
in
the
case
of
“
∞
Ψ
⊥
”].
[Thus,
the
subgroup
of
units
of
“Ψ
⊥
”
consists
of
the
2l-torsion
subgroup
of
“Ψ”,
while
the
subgroup
of
units
of
“
∞
Ψ
⊥
”
contains
the
entire
torsion
subgroup
of
“
∞
Ψ”.]
Then,
as
m
ranges
over
the
elements
of
Z,
the
actions,
via
the
[relevant]
Kummer
isomorphisms
of
(i),
of
the
various
±ell
±ell
n,m
n,m
HT
Θ
NF
)
v
(⊆
∞
Ψ
⊥
HT
Θ
NF
)
v
)
on
the
monoids
Ψ
⊥
F
LGP
(
F
LGP
(
ind-topological
modules
±
±
I
Q
(
S
j+1
,j
F(
n,◦
D
)
v
)
⊆
log(
S
j+1
,j
F(
n,◦
D
)
v
)
[where
j
=
1,
.
.
.
,
l
]
—
i.e.,
that
arise
from
applying
the
constructions
of
Proposition
3.4,
(ii),
in
the
vertically
coric
context
of
(i)
above
[cf.
also
the
notational
conventions
of
Proposition
3.2,
(ii)]
—
are
mutually
105
106
SHINICHI
MOCHIZUKI
compatible,
relative
to
the
log-links
of
the
n-th
column
of
the
Gaussian
log-theta-lattice
under
consideration,
in
the
sense
that
the
only
portions
of
these
actions
that
are
possibly
related
to
one
another
via
these
log-links
are
the
indeterminacies
with
respect
to
multiplication
by
roots
of
unity
in
the
domains
of
the
log-links,
that
is
to
say,
indeterminacies
at
m
that
correspond,
via
the
log-link,
to
“addition
by
zero”
—
i.e.,
to
no
indeterminacy!
—
at
m
+
1.
Now
let
us
think
of
the
submodules
of
Galois
invariants
[cf.
the
discussion
of
Proposition
3.4,
(ii)]
of
the
various
groups
of
units,
for
v
∈
V,
(Ψ
cns
(
n,m
F
)
|t|
)
×
v
,
Ψ
F
LGP
(
n,m
HT
Θ
±ell
NF
×
)
v
and
the
splitting
monoids,
for
v
∈
V
bad
,
n,m
Ψ
⊥
HT
Θ
F
LGP
(
±ell
NF
)
v
as
acting
on
various
portions
of
the
modules,
for
v
Q
∈
V
Q
,
±
I
Q
(
S
j+1
F(
n,◦
D
)
v
Q
)
not
via
a
single
Kummer
isomorphism
as
in
(i)
—
which
fails
to
be
com-
patible
with
the
log-links
of
the
Gaussian
log-theta-lattice!
—
but
rather
via
the
totality
of
the
various
pre-composites
of
Kummer
isomorphisms
with
iterates
[cf.
Remark
1.1.1]
of
the
log-links
of
the
Gaussian
log-theta-lattice
—
i.e.,
precisely
as
was
described
in
detail
in
(a),
(b),
(c)
above
[cf.
also
the
discussion
of
Remark
3.11.4
below].
Thus,
one
obtains
a
sort
of
“log-Kummer
correspondence”
be-
tween
the
totality,
as
m
ranges
over
the
elements
of
Z,
of
the
various
groups
of
units
and
splitting
monoids
just
discussed
[i.e.,
which
are
labeled
by
“n,
m”]
and
their
actions
[as
just
described]
on
the
“I
Q
”
labeled
by
“n,
◦”
which
is
invariant
with
respect
to
the
translation
symmetries
[cf.
Proposition
1.3,
(iv)]
of
the
n-th
column
of
the
Gaussian
log-theta-lattice
[cf.
the
discussion
of
Remark
1.2.2,
(iii)].
Proof.
The
various
assertions
of
Proposition
3.5
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Example
3.6.
Concrete
Representations
of
Global
Frobenioids.
Before
”
of
[IUTchI],
proceeding,
we
pause
to
take
a
closer
look
at
the
Frobenioid
“
†
F
mod
Example
5.1,
(iii),
i.e.,
more
concretely
speaking,
the
Frobenioid
of
arithmetic
line
bundles
on
the
stack
“S
mod
”
of
[IUTchI],
Remark
3.1.5.
Let
us
write
F
mod
for
the
Frobenioid
“
†
F
mod
”
of
[IUTchI],
Example
5.1,
(iii),
in
the
case
where
the
data
denoted
by
the
label
“†”
arises
[in
the
evident
sense]
from
data
as
discussed
in
[IUTchI],
Definition
3.1.
In
the
following
discussion,
we
shall
use
the
notation
of
[IUTchI],
Definition
3.1.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
107
(i)
(Rational
Function
Torsor
Version)
For
each
v
∈
V,
the
valuation
on
×
×
K
v
determined
by
v
determines
a
group
homomorphism
β
v
:
F
mod
→
K
v
×
/O
K
[cf.
v
Remark
3.6.1
below].
Then
let
us
define
a
category
F
MOD
as
follows.
An
object
T
=
(T,
{t
v
}
v∈V
)
of
F
MOD
consists
of
a
collection
of
data
×
(a)
an
F
mod
-torsor
T
;
(b)
for
each
v
∈
V,
a
trivalization
t
v
of
the
torsor
T
v
obtained
from
T
by
executing
the
“change
of
structure
group”
operation
determined
by
the
homomorphism
β
v
subject
to
the
condition
that
there
exists
an
element
t
∈
T
such
that
t
v
coincides
with
the
trivialization
of
T
v
determined
by
t
for
all
but
finitely
many
v.
An
ele-
mentary
morphism
T
1
=
(T
1
,
{t
1,v
}
v∈V
)
→
T
2
=
(T
2
,
{t
2,v
}
v∈V
)
between
objects
of
∼
×
F
MOD
is
defined
to
be
an
isomorphism
T
1
→
T
2
of
F
mod
-torsors
which
is
integral
-orbit
at
each
v
∈
V,
i.e.,
maps
the
trivialization
t
1,v
to
an
element
of
the
O
K
v
of
t
2,v
.
There
is
an
evident
notion
of
composition
of
elementary
morphisms,
as
well
as
an
evident
notion
of
tensor
powers
T
⊗n
,
for
n
∈
Z,
of
an
object
T
of
.
A
morphism
T
1
=
(T
1
,
{t
1,v
}
v∈V
)
→
T
2
=
(T
2
,
{t
2,v
}
v∈V
)
between
objects
F
MOD
of
F
MOD
is
defined
to
consist
of
a
positive
integer
n
and
an
elementary
morphism
(T
1
)
⊗n
→
T
2
.
There
is
an
evident
notion
of
composition
of
morphisms.
Thus,
F
MOD
forms
a
category.
In
fact,
one
verifies
immediately
that,
from
the
point
of
admits
a
nat-
view
of
the
theory
of
Frobenioids
developed
in
[FrdI],
[FrdII],
F
MOD
ural
Frobenioid
structure
[cf.
[FrdI],
Definition
1.3],
for
which
the
base
category
is
the
category
with
precisely
one
arrow.
Relative
to
this
Frobenioid
structure,
the
elementary
morphisms
are
precisely
the
linear
morphisms,
and
the
positive
integer
is
the
Frobenius
degree
“n”
that
appears
in
the
definition
of
a
morphism
of
F
MOD
of
the
morphism.
Moreover,
by
associating
to
an
arithmetic
line
bundle
on
S
mod
×
the
F
mod
-torsor
determined
by
restricting
the
line
bundle
to
the
generic
point
of
S
mod
and
the
local
trivializations
at
v
∈
V
determined
by
the
various
local
inte-
gral
structures,
one
verifies
immediately
that
there
exists
a
natural
isomorphism
of
Frobenioids
∼
→
F
MOD
F
mod
×
×
that
induces
the
identity
morphism
F
mod
→
F
mod
on
the
associated
rational
func-
tion
monoids
[cf.
[FrdI],
Corollary
4.10].
as
(ii)
(Local
Fractional
Ideal
Version)
Let
us
define
a
category
F
mod
follows.
An
object
J
=
{J
v
}
v∈V
of
F
mod
consists
of
a
collection
of
“fractional
ideals”
J
v
⊆
K
v
for
each
v
∈
V
—
i.e.,
a
finitely
generated
nonzero
O
K
v
-submodule
of
K
v
when
v
∈
V
non
;
a
positive
real
def
multiple
of
O
K
v
=
{λ
∈
K
v
|
|λ|
≤
1}
⊆
K
v
when
v
∈
V
arc
—
such
that
J
v
=
O
K
v
for
all
but
finitely
many
v.
If
J
=
{J
v
}
v∈V
is
an
object
of
F
mod
,
then
for
any
element
×
f
∈
F
mod
,
one
obtains
an
object
f
·
J
=
{f
·
J
v
}
v∈V
of
F
mod
by
multiplying
each
of
the
fractional
ideals
J
v
by
f
.
Moreover,
if
J
=
{J
v
}
v∈V
is
an
object
of
F
mod
,
then
⊗n
for
any
n
∈
Z,
there
is
an
evident
notion
of
the
n-th
tensor
power
J
of
J
.
An
is
elementary
morphism
J
1
=
{J
1,v
}
v∈V
→
J
2
=
{J
2,v
}
v∈V
between
objects
of
F
mod
108
SHINICHI
MOCHIZUKI
×
defined
to
be
an
element
f
∈
F
mod
that
is
integral
with
respect
to
J
1
and
J
2
in
the
sense
that
f
·
J
1,v
⊆
J
2,v
for
each
v
∈
V.
There
is
an
evident
notion
of
composition
of
elementary
morphisms.
A
morphism
J
1
=
{J
1,v
}
v∈V
→
J
2
=
{J
2,v
}
v∈V
between
is
defined
to
consist
of
a
positive
integer
n
and
an
elementary
objects
of
F
mod
⊗n
morphism
(J
1
)
→
J
2
.
There
is
an
evident
notion
of
composition
of
morphisms.
Thus,
F
mod
forms
a
category.
In
fact,
one
verifies
immediately
that,
from
the
point
of
view
of
the
theory
of
Frobenioids
developed
in
[FrdI],
[FrdII],
F
mod
admits
a
natural
Frobenioid
structure
[cf.
[FrdI],
Definition
1.3],
for
which
the
base
category
is
the
category
with
precisely
one
arrow.
Relative
to
this
Frobenioid
structure,
the
elementary
morphisms
are
precisely
the
linear
morphisms,
and
the
positive
integer
is
the
Frobenius
degree
“n”
that
appears
in
the
definition
of
a
morphism
of
F
mod
of
the
morphism.
Moreover,
by
associating
to
an
object
J
=
{J
v
}
v∈V
of
F
mod
the
arithmetic
line
bundle
on
S
mod
obtained
from
the
trivial
arithmetic
line
bundle
on
S
mod
by
modifying
the
integral
structure
of
the
trivial
line
bundle
at
v
∈
V
in
the
fashion
prescribed
by
J
v
,
one
verifies
immediately
that
there
exists
a
natural
isomorphism
of
Frobenioids
∼
→
F
mod
F
mod
×
×
that
induces
the
identity
morphism
F
mod
→
F
mod
on
the
associated
rational
func-
tion
monoids
[cf.
[FrdI],
Corollary
4.10].
(iii)
By
composing
the
isomorphisms
of
Frobenioids
of
(i)
and
(ii),
one
thus
obtains
a
natural
isomorphism
of
Frobenioids
∼
→
F
MOD
F
mod
×
×
→
F
mod
on
the
associated
rational
func-
that
induces
the
identity
morphism
F
mod
tion
monoids
[cf.
[FrdI],
Corollary
4.10].
One
verifies
immediately
that
although
the
above
isomorphism
of
Frobenioids
is
not
necessarily
determined
by
the
condition
×
,
the
induced
isomorphism
between
that
it
induce
the
identity
morphism
on
F
mod
the
respective
perfections
[hence
also
on
realifications]
of
F
mod
,
F
MOD
is
completely
determined
by
this
condition.
Remark
3.6.1.
Note
that,
as
far
as
the
various
constructions
of
Example
3.6,
(i),
are
concerned,
the
various
homomorphisms
β
v
,
for
v
∈
V,
may
be
thought
of,
alternatively,
as
a
collection
of
×
×
subquotients
of
the
perfection
(F
mod
)
pf
of
F
mod
—
each
of
which
is
equipped
with
a
submonoid
of
“nonnegative
elements”
—
that
are
completely
determined
by
the
ring
structure
of
the
field
F
mod
[i.e.,
equipped
with
its
structure
as
the
field
of
moduli
of
X
F
].
Remark
3.6.2.
(i)
In
the
theory
to
be
developed
below,
we
shall
be
interested
in
relating
certain
Frobenioids
—
which
will,
in
fact,
be
isomorphic
to
the
realification
of
F
mod
—
that
lie
on
opposite
sides
of
[a
certain
enhanced
version
of]
the
Θ
×μ
gau
-link
to
one
another.
In
particular,
at
the
level
of
objects
of
the
Frobenioids
involved,
it
only
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
109
makes
sense
to
work
with
isomorphism
classes
of
objects
that
are
preserved
by
the
isomorphisms
of
Frobenioids
that
appear.
Here,
we
note
that
the
isomorphism
classes
of
the
sort
of
Frobenioids
that
appear
in
this
context
are
determined
by
the
divisor
and
rational
function
monoids
of
the
[model]
Frobenioid
in
question
[cf.
the
constructions
given
in
[FrdI],
Theorem
5.2,
(i),
(ii)].
In
this
context,
we
observe
×
of
F
mod
satisfies
the
following
fundamental
that
the
rational
function
monoid
F
mod
property:
×
admits
a
natural
additive
structure.
[the
union
with
{0}
of]
F
mod
In
this
context,
we
note
that
this
property
is
not
satisfied
by
(a)
the
rational
function
monoids
of
the
perfection
or
realification
of
F
mod
×
(b)
subgroups
Γ
⊆
F
mod
—
such
as,
for
instance,
the
trivial
subgroup
{1}
or
the
subgroup
of
S-units,
for
S
⊆
V
mod
a
nonempty
finite
subset
—
that
do
not
arise
as
the
multiplicative
group
of
some
subfield
of
F
mod
[cf.
[AbsTopIII],
Remark
5.10.2,
(iv)].
The
significance
of
this
fundamental
property
is
that
it
allows
one
to
represent
the
additively,
i.e.,
as
modules
—
cf.
the
point
of
view
of
Example
3.6,
objects
of
F
mod
(ii).
At
a
more
concrete
level,
if,
in
the
notation
of
(b),
one
considers
the
result
of
“adding”
two
elements
of
a
Γ-torsor
[cf.
the
point
of
view
of
Example
3.6,
(i)!],
then
the
resulting
“sum”
can
only
be
rendered
meaningful,
relative
to
the
given
Γ-torsor,
will
be
if
Γ
is
additively
closed.
The
additive
representation
of
objects
of
F
mod
of
crucial
importance
in
the
theory
of
the
present
series
of
papers
since
it
will
allow
on
opposite
sides
of
[a
certain
enhanced
version
of]
us
to
relate
objects
of
F
mod
×μ
the
Θ
gau
-link
to
one
another
—
which,
a
priori,
are
only
related
to
one
another
at
the
level
of
realifications
in
a
multiplicative
fashion
—
by
means
of
[“additive”]
mono-analytic
log-shells
[cf.
the
discussion
of
[IUTchII],
Remark
4.7.2].
(ii)
One
way
to
understand
the
content
of
the
discussion
of
(i)
is
as
follows:
whereas
×
the
construction
of
F
mod
depends
on
the
additive
structure
of
F
mod
in
an
essential
way,
is
strictly
multiplicative
in
nature.
the
construction
of
F
MOD
Indeed,
the
construction
of
F
MOD
given
in
Example
3.6,
(i),
is
essentially
the
same
as
the
construction
of
F
mod
given
in
[FrdI],
Example
6.3
[i.e.,
in
effect,
in
[FrdI],
Theorem
5.2,
(i)].
From
this
point
of
view,
it
is
natural
to
identify
F
MOD
with
F
mod
via
the
natural
isomorphism
of
Frobenioids
of
Example
3.6,
(i).
We
shall
often
do
this
in
the
theory
to
be
developed
below.
Proposition
3.7.
(Global
Packet-theoretic
Frobenioids)
(i)
(Single
Packet
Rational
Function
Torsor
Version)
In
the
notation
of
Proposition
3.3:
For
each
α
∈
A,
there
is
an
algorithm
for
constructing,
as
discussed
in
Example
3.6,
(i)
[cf.
also
Remark
3.6.1],
from
the
[number]
field
given
by
the
image
(
†
M
MOD
)
α
110
SHINICHI
MOCHIZUKI
of
the
composite
(
†
M
mod
)
α
→
(
†
M
mod
)
A
→
log(
A
F
V
Q
)
)
α
,
to-
of
the
homomorphisms
of
Proposition
3.3,
(i),
(ii),
a
Frobenioid
(
†
F
MOD
gether
with
a
natural
isomorphism
of
Frobenioids
∼
)
α
→
(
†
F
MOD
)
α
(
†
F
mod
[cf.
the
notation
of
[IUTchII],
Corollary
4.8,
(ii)]
that
induces
the
tautological
∼
†
isomorphism
(
†
M
mod
)
α
→
(
M
MOD
)
α
on
the
associated
rational
function
monoids
[cf.
Example
3.6,
(i)].
We
shall
often
use
this
isomorphism
of
Frobenioids
to
R
identify
(
†
F
mod
)
α
with
(
†
F
MOD
)
α
[cf.
Remark
3.6.2,
(ii)].
Write
(
†
F
MOD
)
α
for
†
the
realification
of
(
F
MOD
)
α
.
(ii)
(Single
Packet
Local
Fractional
Ideal
Version)
In
the
notation
of
Propositions
3.3,
3.4:
For
each
α
∈
A,
there
is
an
algorithm
for
constructing,
as
def
discussed
in
Example
3.6,
(ii),
from
the
[number]
field
(
†
M
mod
)
α
=
(
†
M
MOD
)
α
[cf.
(i)]
and
the
Galois
invariants
of
the
local
monoids
Ψ
log(
A,α
F
v
)
⊆
log(
A,α
F
v
)
for
v
∈
V
of
Proposition
3.4,
(i)
—
i.e.,
so
the
corresponding
local
“fractional
ideal
J
v
”
of
Example
3.6,
(ii),
is
a
subset
[indeed
a
submodule
when
v
∈
V
non
]
of
I
Q
(
A,α
F
v
)
whose
Q-span
is
equal
to
I
Q
(
A,α
F
v
)
[cf.
the
notational
conventions
of
Proposition
3.2,
(ii)]
—
a
Frobenioid
(
†
F
mod
)
α
,
together
with
natural
isomor-
phisms
of
Frobenioids
∼
)
α
→
(
†
F
mod
)
α
;
(
†
F
mod
∼
(
†
F
mod
)
α
→
(
†
F
MOD
)
α
∼
∼
†
†
that
induce
the
tautological
isomorphisms
(
†
M
mod
)
α
→
(
M
mod
)
α
,
(
M
mod
)
α
→
(
†
M
MOD
)
α
on
the
associated
rational
function
monoids
[cf.
the
natural
isomorphism
R
)
α
for
the
realification
of
Frobenioids
of
(i);
Example
3.6,
(ii),
(iii)].
Write
(
†
F
mod
†
of
(
F
mod
)
α
.
(iii)
(Global
Realified
LGP-Frobenioids)
In
the
notation
of
Proposition
∼
3.4:
By
applying
the
composites
of
the
isomorphisms
of
Frobenioids
“
†
C
j
→
∼
R
R
)
j
”
of
[IUTchII],
Corollary
4.8,
(iii),
with
the
realifications
“(
†
F
mod
)
α
→
(
†
F
mod
R
(
†
F
MOD
)
α
”
of
the
isomorphisms
of
Frobenioids
of
(i)
above
to
the
global
realified
Frobenioid
portion
†
C
gau
of
the
F
-prime-strip
†
F
gau
of
[IUTchII],
Corollary
4.10,
(ii)
[cf.
Remarks
1.5.3,
(iii);
3.3.2,
(i)],
one
obtains
a
functorial
algorithm
±ell
in
the
log-link
of
Θ
±ell
NF-Hodge
theaters
‡
HT
Θ
Proposition
3.4,
(ii),
for
constructing
a
Frobenioid
±ell
(
†
HT
Θ
C
LGP
NF
NF
log
−→
†
±ell
HT
Θ
NF
of
)
—
which
we
refer
to
as
a
“global
realified
LGP-Frobenioid”.
Here,
we
note
±ell
that
the
notation
“(
†
HT
Θ
NF
)”
constitutes
a
slight
abuse
of
notation.
In
par-
def
=
C
LGP
(
†
HT
Θ
ticular,
the
global
realified
Frobenioid
†
C
LGP
±ell
NF
),
together
with
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
111
±ell
the
collection
of
data
Ψ
F
LGP
(
†
HT
Θ
NF
)
constructed
in
Proposition
3.4,
(ii),
give
rise,
in
a
natural
fashion,
to
an
F
-prime-strip
†
F
LGP
∼
=
(
†
C
LGP
,
Prime(
†
C
LGP
)
→
V,
†
F
LGP
,
{
†
ρ
LGP,v
}
v∈V
)
—
cf.
the
construction
of
the
F
-prime-strip
†
F
gau
in
[IUTchII],
Corollary
4.10,
(ii)
—
together
with
a
natural
isomorphism
∼
†
F
gau
†
F
LGP
→
of
F
-prime-strips
[i.e.,
that
arises
tautologically
from
the
construction
of
†
F
LGP
!].
(iv)
(Global
Realified
lgp-Frobenioids)
In
the
situation
of
(iii)
above,
write
±ell
±ell
def
def
Ψ
F
lgp
(
†
HT
Θ
NF
)
=
Ψ
F
LGP
(
†
HT
Θ
NF
),
†
F
lgp
=
†
F
LGP
.
Then
by
replacing,
in
∼
R
R
)
α
→
(
†
F
MOD
)
α
”
by
the
natural
the
construction
of
(iii),
the
isomorphisms
“(
†
F
mod
∼
†
R
†
R
isomorphisms
“(
F
mod
)
α
→
(
F
mod
)
α
”
[cf.
(ii)],
one
obtains
a
functorial
algo-
rithm
in
the
log-link
of
Θ
±ell
NF-Hodge
theaters
‡
HT
Θ
of
Proposition
3.4,
(ii),
for
constructing
a
Frobenioid
±ell
†
(
HT
Θ
C
lgp
NF
±ell
NF
log
±ell
−→
†
HT
Θ
NF
)
—
which
we
refer
to
as
a
“global
realified
lgp-Frobenioid”
—
as
well
as
an
F
-prime-strip
†
F
lgp
∼
=
(
†
C
lgp
,
Prime(
†
C
lgp
)
→
V,
†
F
lgp
,
{
†
ρ
lgp,v
}
v∈V
)
±ell
def
†
—
where
we
write
†
C
lgp
=
C
lgp
(
HT
Θ
NF
)
—
together
with
tautological
iso-
morphisms
∼
∼
†
F
gau
→
†
F
→
†
F
LGP
lgp
of
F
-prime-strips
[cf.
(iii)].
(v)
(Realified
Product
Embeddings
and
Non-realified
Global
Frobe-
±ell
±ell
†
(
†
HT
Θ
NF
),
C
lgp
(
HT
Θ
NF
)
given
in
(iii)
nioids)
The
constructions
of
C
LGP
and
(iv)
above
give
rise
to
a
commutative
diagram
of
categories
(
†
HT
Θ
C
LGP
⏐
⏐
†
C
lgp
(
HT
Θ
±ell
±ell
NF
NF
)
→
)
→
R
(
†
F
MOD
)
j
j∈F
l
⏐
⏐
R
(
†
F
mod
)
j
j∈F
l
—
where
the
horizontal
arrows
are
embeddings
that
arise
tautologically
from
the
constructions
of
(iii)
and
(iv)
[cf.
[IUTchII],
Remark
4.8.1,
(i)];
the
vertical
arrows
are
isomorphisms;
the
left-hand
vertical
arrow
arises
from
the
second
isomorphism
that
appears
in
the
final
display
of
(iv);
the
right-hand
vertical
arrow
is
the
product
of
the
realifications
of
copies
of
the
inverse
of
the
second
isomorphism
that
appears
)
j
—
in
the
final
display
of
(ii).
In
particular,
by
applying
the
definition
of
(
†
F
mod
112
SHINICHI
MOCHIZUKI
i.e.,
in
terms
of
local
fractional
ideals
[cf.
(ii)]
—
together
with
the
products
of
realification
functors
(
†
F
mod
)
j
→
j∈F
l
R
(
†
F
mod
)
j
j∈F
l
[cf.
[FrdI],
Proposition
5.3],
one
obtains
an
algorithm
for
constructing,
in
a
fash-
ion
compatible
[in
the
evident
sense]
with
the
local
isomorphisms
{
†
ρ
lgp,v
}
v∈V
,
±ell
†
(
HT
Θ
{
†
ρ
LGP,v
}
v∈V
of
(iii)
and
(iv),
objects
of
the
[global!]
categories
C
lgp
NF
),
Θ
±ell
NF
C
LGP
(
†
HT
)
from
the
local
fractional
ideals
generated
by
elements
of
the
monoids
[cf.
(iv);
Proposition
3.4,
(ii)]
Ψ
F
lgp
(
†
HT
Θ
±ell
NF
)
v
for
v
∈
V
bad
.
Proof.
The
various
assertions
of
Proposition
3.7
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Definition
3.8.
def
⊥
(i)
In
the
situation
of
Proposition
3.7,
(iv),
(v),
write
Ψ
⊥
F
lgp
(−)
v
=
Ψ
F
LGP
(−)
v
,
for
v
∈
V
bad
[cf.
the
notation
of
Proposition
3.5,
(ii),
(c)].
Then
we
shall
refer
to
the
object
of
(
†
F
MOD
)
j
or
(
†
F
mod
)
j
j∈F
l
j∈F
l
±ell
—
as
well
as
its
realification,
regarded
as
an
object
of
†
C
LGP
=
C
LGP
(
†
HT
Θ
NF
)
±ell
†
=
C
lgp
(
HT
Θ
NF
)
[cf.
Proposition
3.7,
(iii),
(iv),
(v)]
—
determined
by
or
†
C
lgp
any
collection,
indexed
by
v
∈
V
bad
,
of
generators
up
to
torsion
of
the
monoids
Θ
±ell
NF
†
Ψ
⊥
)
v
as
a
Θ-pilot
object
[cf.
also
Remark
3.8.1
below].
We
shall
F
lgp
(
HT
refer
to
the
object
of
the
[global
realified]
Frobenioid
†
C
of
[IUTchII],
Corollary
4.10,
(i),
determined
by
any
collection,
indexed
by
v
∈
V
bad
,
of
generators
up
to
torsion
of
the
splitting
monoid
associated
to
the
split
Frobenioid
†
F
,v
[i.e.,
the
data
indexed
by
v
of
the
F
-prime-strip
†
F
of
[IUTchII],
Corollary
4.10,
(i)]
—
that
is
to
say,
at
a
more
concrete
level,
determined
by
the
“q
”,
for
v
v
∈
V
[cf.
the
notation
of
[IUTchI],
Example
3.2,
(iv)]
—
as
a
q-pilot
object
[cf.
also
Remark
3.8.1
below].
bad
(ii)
Let
‡
HT
Θ
±ell
NF
−→
†
HT
Θ
log
±ell
NF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
113
be
a
log-link
of
Θ
±ell
NF-Hodge
theaters
and
∗
±ell
HT
Θ
NF
a
Θ
±ell
NF-Hodge
theater
[all
relative
to
the
given
initial
Θ-data].
Recall
the
F
-
prime-strip
∗
F
±ell
constructed
from
∗
HT
Θ
NF
in
[IUTchII],
Corollary
4.10,
(i).
Following
the
nota-
tional
conventions
of
[IUTchII],
Corollary
4.10,
(iii),
let
us
write
∗
F
×μ
(respec-
†
×μ
)
for
the
F
×μ
-prime-strip
associated
to
the
F
-prime-
tively,
†
F
×μ
LGP
;
F
lgp
†
strip
∗
F
(respectively,
†
F
LGP
;
F
lgp
)
[cf.
Proposition
3.7,
(iii),
(iv);
[IUTchII],
Definition
4.9,
(viii);
the
functorial
algorithm
described
in
[IUTchII],
Definition
4.9,
(vi)].
Then
—
in
the
style
of
[IUTchII],
Corollary
4.10,
(iii)
—
we
shall
refer
×μ
∼
∗
×μ
→
F
as
the
to
the
full
poly-isomorphism
of
F
×μ
-prime-strips
†
F
LGP
×μ
Θ
LGP
-link
†
HT
Θ
±ell
±ell
NF
Θ
×μ
LGP
−→
∗
HT
Θ
±ell
NF
±ell
±ell
log
±ell
log
±ell
from
†
HT
Θ
NF
to
∗
HT
Θ
NF
,
relative
to
the
log-link
‡
HT
Θ
NF
−→
†
HT
Θ
NF
,
∼
and
to
the
full
poly-isomorphism
of
F
×μ
-prime-strips
†
F
×μ
→
∗
F
×μ
as
lgp
the
Θ
×μ
lgp
-link
†
from
†
HT
Θ
±ell
NF
±ell
HT
Θ
to
∗
HT
Θ
±ell
NF
NF
Θ
×μ
lgp
−→
∗
HT
Θ
±ell
NF
,
relative
to
the
log-link
‡
HT
Θ
±ell
NF
−→
†
HT
Θ
±ell
NF
(iii)
Let
{
n,m
HT
Θ
NF
}
n,m∈Z
be
a
collection
of
distinct
Θ
±ell
NF-Hodge
the-
aters
[relative
to
the
given
initial
Θ-data]
indexed
by
pairs
of
integers.
Then
we
shall
refer
to
the
first
(respectively,
second)
diagram
..
.
⏐
log
⏐
...
...
Θ
×μ
LGP
n,m+1
Θ
×μ
LGP
n,m
−→
−→
±ell
HT
Θ
⏐
log
⏐
±ell
..
.
⏐
log
⏐
NF
HT
Θ
NF
⏐
log
⏐
..
.
Θ
×μ
LGP
n+1,m+1
Θ
×μ
LGP
n+1,m
−→
−→
HT
Θ
⏐
log
⏐
±ell
±ell
HT
Θ
⏐
log
⏐
..
.
NF
NF
Θ
×μ
LGP
−→
Θ
×μ
LGP
−→
...
...
.
114
SHINICHI
MOCHIZUKI
..
.
⏐
log
⏐
...
...
Θ
×μ
lgp
−→
Θ
×μ
lgp
−→
n,m+1
n,m
..
.
⏐
log
⏐
±ell
HT
Θ
⏐
log
⏐
Θ
×μ
lgp
−→
NF
n+1,m+1
Θ
×μ
lgp
Θ
±ell
NF
HT
⏐
log
⏐
−→
n+1,m
±ell
HT
Θ
⏐
log
⏐
NF
Θ
±ell
NF
HT
⏐
log
⏐
..
.
Θ
×μ
lgp
−→
Θ
×μ
lgp
−→
...
...
..
.
—
where
the
vertical
arrows
are
the
full
log-links,
and
the
horizontal
arrow
of
the
±ell
±ell
first
(respectively,
second)
diagram
from
n,m
HT
Θ
NF
to
n+1,m
HT
Θ
NF
is
the
±ell
±ell
×μ
n,m
Θ
×μ
HT
Θ
NF
to
n+1,m
HT
Θ
NF
,
relative
LGP
-
(respectively,
Θ
lgp
-)
link
from
±ell
±ell
log
to
the
full
log-link
n,m−1
HT
Θ
NF
−→
n,m
HT
Θ
NF
[cf.
(ii)]
—
as
the
[LGP-
Gaussian]
(respectively,
[lgp-Gaussian])
log-theta-lattice.
Thus,
[cf.
Definition
1.4]
either
of
these
diagrams
may
be
represented
symbolically
by
an
oriented
graph
..
..
.
.
⏐
⏐
⏐
⏐
...
−→
•
⏐
⏐
−→
•
⏐
⏐
−→
.
.
.
...
−→
•
⏐
⏐
−→
•
⏐
⏐
−→
.
.
.
..
.
..
.
±ell
—
where
the
“•’s”
correspond
to
the
“
n,m
HT
Θ
NF
”.
Remark
3.8.1.
The
LGP-Gaussian
and
lgp-Gaussian
log-theta-lattices
are,
of
course,
closely
related,
but,
in
the
theory
to
be
developed
below,
we
shall
mainly
be
interested
in
the
LGP-Gaussian
log-theta-lattice
[for
reasons
to
be
explained
in
Remark
3.10.1,
(ii),
below].
On
the
other
hand,
our
computation
of
the
Θ
×μ
LGP
-link
×μ
will
involve
the
Θ
lgp
-link,
as
well
as
related
Θ-pilot
objects,
in
an
essential
way.
×μ
Here,
we
note,
for
future
reference,
that
both
the
Θ
×μ
LGP
-
and
the
Θ
lgp
-link
map
Θ-pilot
objects
to
q-pilot
objects.
Also,
we
observe
that
this
terminology
of
“Θ-
pilot/q-pilot
objects”
is
consistent
with
the
notion
of
a
“pilot
object”
associated
to
a
F
×μ
-prime-strip,
as
defined
in
[IUTchII],
Definition
4.9,
(viii).
Remark
3.8.2.
One
verifies
immediately
that
the
main
results
obtained
so
far
concerning
Gaussian
log-theta-lattices
—
namely,
Theorem
1.5,
Proposition
2.1,
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
115
Corollary
2.3
[cf.
also
Remark
2.3.2],
and
Proposition
3.5
—
generalize
immediately
[indeed,
“formally”]
to
the
case
of
LGP-
or
lgp-Gaussian
log-theta-lattices.
Indeed,
the
substantive
content
of
these
results
concerns
portions
of
the
log-theta-
lattices
involved
that
are
substantively
unaffected
by
the
transition
from
“Gaussian”
to
“LGP-
or
lgp-Gaussian”.
Remark
3.8.3.
In
the
definition
of
the
various
horizontal
arrows
of
the
log-
theta-lattices
discussed
in
Definition
3.8,
(iii),
it
may
appear
to
the
reader,
at
first
glance,
that,
instead
of
working
with
F
×μ
-prime-strips,
it
might
in
fact
be
sufficient
to
replace
the
unit
[i.e.,
F
×μ
-prime-strip]
portions
of
these
prime-
strips
by
the
associated
log-shells
[cf.
Proposition
1.2,
(vi),
(vii)],
on
which,
at
nonarchimedean
v
∈
V,
the
associated
local
Galois
groups
act
trivially.
In
fact,
however,
this
is
not
the
case.
That
is
to
say,
the
nontrivial
Galois
action
on
the
local
unit
portions
of
the
F
×μ
-prime-strips
involved
is
necessary
in
order
to
consider
the
Kummer
theory
[cf.
Proposition
3.5,
(i),
(ii),
as
well
as
Proposition
3.10,
(i),
(iii);
Theorem
3.11,
(iii),
(c),
(d),
below]
of
the
various
local
and
global
objects
for
which
the
log-shells
serve
as
“multiradial
containers”
[cf.
the
discussion
of
Remark
1.5.2].
Here,
we
recall
that
this
Kummer
theory
plays
a
crucial
role
in
the
theory
of
the
present
series
of
papers
in
relating
corresponding
Frobenius-like
and
étale-like
objects
[cf.
the
discussion
of
Remark
1.5.4,
(i)].
Proposition
3.9.
(Log-volume
for
Packets
and
Processions)
(i)
(Local
Holomorphic
Packets)
In
the
situation
of
Proposition
3.2,
(i),
(ii):
Suppose
that
V
v
|
v
Q
∈
V
non
Q
,
α
∈
A.
Then
the
p
v
Q
-adic
log-volume
on
each
of
the
direct
summand
p
v
Q
-adic
fields
of
I
Q
(
α
F
v
Q
),
I
Q
(
A
F
v
Q
),
and
I
Q
(
A,α
F
v
)
—
cf.
the
direct
sum
decompositions
of
Proposition
3.1,
(i),
together
with
the
discussion
of
normalized
weights
in
Remark
3.1.1,
(ii),
(iii),
(iv)
—
determines
[cf.
[AbsTopIII],
Proposition
5.7,
(i)]
log-volumes
Q
α
μ
log
α,v
Q
:
M(I
(
F
v
Q
))
→
R;
Q
A
μ
log
A,v
Q
:
M(I
(
F
v
Q
))
→
R
Q
A,α
F
v
))
→
R
μ
log
A,α,v
:
M(I
(
—
where
we
write
“M(−)”
for
the
set
of
nonempty
compact
open
subsets
of
“(−)”
—
such
that
the
log-volume
of
each
of
the
“local
holomorphic”
integral
structures
of
Proposition
3.1,
(ii)
—
i.e.,
the
elements
O
α
F
v
Q
⊆
I
Q
(
α
F
v
Q
);
O
A
F
v
Q
⊆
I
Q
(
A
F
v
Q
);
O
A,α
F
v
⊆
I
Q
(
A,α
F
v
)
of
“M(−)”
given
by
the
integral
structures
discussed
in
Proposition
3.1,
(ii),
on
each
of
the
direct
summand
p
v
Q
-adic
fields
—
is
equal
to
zero.
Here,
we
assume
that
these
log-volumes
are
normalized
so
that
multiplication
of
an
element
of
“M(−)”
by
p
v
corresponds
to
adding
the
quantity
−log(p
v
)
∈
R;
we
shall
refer
to
this
nor-
malization
as
the
packet-normalization.
Suppose
that
V
v
|
v
Q
∈
V
arc
Q
,
α
∈
A.
Then
the
sum
of
the
radial
log-volumes
on
each
of
the
direct
summand
complex
archimedean
fields
of
I
Q
(
α
F
v
Q
),
I
Q
(
A
F
v
Q
),
and
I
Q
(
A,α
F
v
)
—
cf.
the
direct
sum
decompositions
of
Proposition
3.1,
(i),
together
with
the
discussion
of
normalized
116
SHINICHI
MOCHIZUKI
weights
in
Remark
3.1.1,
(ii),
(iii),
(iv)
—
determines
[cf.
[AbsTopIII],
Proposi-
tion
5.7,
(ii)]
log-volumes
Q
α
μ
log
α,v
Q
:
M(I
(
F
v
Q
))
→
R;
Q
A
μ
log
A,v
Q
:
M(I
(
F
v
Q
))
→
R
Q
A,α
μ
log
F
v
))
→
R
A,α,v
:
M(I
(
—
where
we
write
“M(−)”
for
the
set
of
compact
closures
of
nonempty
open
subsets
of
“(−)”
—
such
that
the
log-volume
of
each
of
the
“local
holomorphic”
integral
structures
of
Proposition
3.1,
(ii)
—
i.e.,
the
elements
O
α
F
v
Q
⊆
I
Q
(
α
F
v
Q
);
O
A
F
v
Q
⊆
I
Q
(
A
F
v
Q
);
O
A,α
F
v
⊆
I
Q
(
A,α
F
v
)
of
“M(−)”
given
by
the
products
of
the
integral
structures
discussed
in
Proposition
3.1,
(ii),
on
each
of
the
direct
summand
complex
archimedean
fields
—
is
equal
to
zero.
Here,
we
assume
that
these
log-volumes
are
normalized
so
that
multiplication
of
an
element
of
“M(−)”
by
e
=
2.71828
.
.
.
corresponds
to
adding
the
quantity
1
=
log(e)
∈
R;
we
shall
refer
to
this
normalization
as
the
packet-normalization.
In
both
the
nonarchimedean
and
archimedean
cases,
“μ
log
A,v
Q
”
is
invariant
with
respect
to
permutations
of
A.
Finally,
when
working
with
collections
of
capsules
in
a
procession,
as
in
Proposition
3.4,
(ii),
we
obtain,
in
both
the
nonarchimedean
and
archimedean
cases,
log-volumes
on
the
products
of
the
“M(−)”
associated
to
the
various
capsules
under
consideration,
which
we
normalize
by
taking
the
average,
over
the
various
capsules
under
consideration;
we
shall
refer
to
this
normalization
as
the
procession-normalization
[cf.
Remark
3.9.3
below].
(ii)
(Mono-analytic
Compatibility)
In
the
situation
of
Proposition
3.2,
(i),
(ii):
Suppose
that
V
v
|
v
Q
∈
V
Q
.
Then
by
applying
the
p
v
Q
-adic
log-volume,
arc
when
v
Q
∈
V
non
Q
,
or
the
radial
log-volume,
when
v
Q
∈
V
Q
,
on
the
mono-analytic
log-shells
“I
†
D
v
”
of
Proposition
1.2,
(vi),
(vii),
(viii),
and
adjusting
appropriately
[cf.
Remark
3.9.1
below
for
more
details]
to
account
for
the
discrepancy
between
the
“local
holomorphic”
integral
structures
of
Proposition
3.1,
(ii),
and
the
“mono-analytic”
integral
structures
of
Proposition
3.2,
(ii),
one
obtains
[by
a
slight
abuse
of
notation]
log-volumes
Q
α
μ
log
α,v
Q
:
M(I
(
D
v
Q
))
→
R;
Q
A
μ
log
A,v
Q
:
M(I
(
D
v
Q
))
→
R
Q
A,α
μ
log
D
v
))
→
R
A,α,v
:
M(I
(
—
where
“M(−)”
is
as
in
(i)
above
—
which
are
compatible
with
the
log-volumes
obtained
in
(i),
relative
to
the
natural
poly-isomorphisms
of
Proposition
3.2,
(i).
In
particular,
these
log-volumes
may
be
constructed
via
a
functorial
algo-
rithm
from
the
D
-prime-strips
under
consideration.
If
one
considers
the
mono-
analyticization
[cf.
[IUTchI],
Proposition
6.9,
(ii)]
of
a
holomorphic
procession
as
in
Proposition
3.4,
(ii),
then
taking
the
average,
as
in
(i)
above,
of
the
packet-
normalized
log-volumes
of
the
above
display
gives
rise
to
procession-normalized
log-volumes,
which
are
compatible,
relative
to
the
natural
poly-isomorphisms
of
Proposition
3.2,
(i),
with
the
procession-normalized
log-volumes
of
(i).
Finally,
by
replacing
“D
”
by
“F
×μ
”
[cf.
also
the
discussion
of
Proposition
1.2,
(vi),
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
117
(vii),
(viii)],
one
obtains
a
similar
theory
of
log-volumes
for
the
various
objects
as-
sociated
to
the
mono-analytic
log-shells
“I
†
F
v
×μ
”,
which
is
compatible
with
the
theory
obtained
for
“D
”
relative
to
the
various
natural
poly-isomorphisms
of
Proposition
3.2,
(i).
(iii)
(Global
Compatibility)
In
the
situation
of
Proposition
3.7,
(i),
(ii):
Write
def
I
Q
(
A
F
v
Q
)
⊆
log(
A
F
V
Q
)
=
log(
A
F
v
Q
)
I
Q
(
A
F
V
Q
)
=
v
Q
∈V
Q
and
v
Q
∈V
Q
M(I
Q
(
A
F
V
Q
))
⊆
M(I
Q
(
A
F
v
Q
))
v
Q
∈V
Q
for
the
subset
of
elements
whose
components,
indexed
by
v
Q
∈
V
Q
,
have
zero
log-
volume
[cf.
(i)]
for
all
but
finitely
many
v
Q
∈
V
Q
.
Then,
by
adding
the
log-volumes
of
(i)
[all
but
finitely
many
of
which
are
zero!]
at
the
various
v
Q
∈
V
Q
,
one
obtains
a
global
log-volume
Q
A
μ
log
A,V
Q
:
M(I
(
F
V
Q
))
→
R
which
is
invariant
with
respect
to
multiplication
by
elements
of
†
Q
A
(
†
M
mod
)
α
=
(
M
MOD
)
α
⊆
I
(
F
V
Q
)
as
well
as
with
respect
to
permutations
of
A,
and,
moreover,
satisfies
the
fol-
lowing
property
concerning
[the
elements
of
“M(−)”
determined
by]
objects
“J
=
)
α
[cf.
Example
3.6,
(ii);
Proposition
3.7,
(ii)]:
the
global
{J
v
}
v∈V
”
of
(
†
F
mod
log
log-volume
μ
A,V
Q
(J
)
is
equal
to
the
degree
of
the
arithmetic
line
bundle
de-
termined
by
J
[cf.
the
discussion
of
Example
3.6,
(ii);
the
natural
isomorphism
∼
(
†
F
mod
)
α
→
(
†
F
mod
)
α
of
Proposition
3.7,
(ii)],
relative
to
a
suitable
normal-
ization.
±ell
(iv)
(log-link
Compatibility)
Let
{
n,m
HT
Θ
NF
}
n,m∈Z
be
a
collection
of
distinct
Θ
±ell
NF-Hodge
theaters
[relative
to
the
given
initial
Θ-data]
—
which
we
think
of
as
arising
from
an
LGP-Gaussian
log-theta-lattice
[cf.
Definition
3.8,
(iii)].
Then
[cf.
also
the
discussion
of
Remark
3.9.4
below]:
(a)
For
n,
m
∈
Z,
the
log-volumes
constructed
in
(i),
(ii),
(iii)
above
deter-
mine
log-volumes
on
the
various
“I
Q
((−))”
that
appear
in
the
construc-
tion
of
the
local/global
LGP-/lgp-monoids/Frobenioids
that
appear
n,m
F
lgp
constructed
in
Proposition
3.7,
in
the
F
-prime-strips
n,m
F
LGP
,
(iii),
(iv),
relative
to
the
log-link
n,m−1
HT
Θ
±ell
NF
log
−→
n,m
±ell
HT
Θ
NF
.
(b)
At
the
level
of
the
Q-spans
of
log-shells
“I
Q
((−))”
that
arise
from
the
various
F-prime-strips
involved,
the
log-volumes
of
(a)
indexed
by
(n,
m)
are
compatible
—
in
the
sense
discussed
in
Propositions
1.2,
(iii);
1.3,
(iii)
—
with
the
corresponding
log-volumes
indexed
by
(n,
m
−
1),
relative
to
the
log-link
n,m−1
HT
Θ
±ell
NF
log
−→
n,m
HT
Θ
±ell
NF
.
118
SHINICHI
MOCHIZUKI
Proof.
The
various
assertions
of
Proposition
3.9
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
3.9.1.
In
the
spirit
of
the
explicit
descriptions
of
Remark
3.1.1,
(i)
[cf.
also
Remark
1.2.2,
(i),
(ii)],
we
make
the
following
observations.
(i)
Suppose
that
v
Q
∈
V
non
Q
.
Write
{v
1
,
.
.
.
,
v
n
v
}
for
the
[distinct!]
elements
Q
def
of
V
that
lie
over
v
Q
.
For
each
i
=
1,
.
.
.
,
n
v
Q
,
set
k
i
=
K
v
i
;
write
O
k
i
⊆
k
i
for
the
ring
of
integers
of
k
i
,
I
i
=
(p
∗
v
Q
)
−1
·
log
k
i
(O
k
×
i
)
⊆
k
i
def
—
where
p
∗
v
Q
=
p
v
if
p
v
Q
is
odd,
p
∗
v
Q
=
p
2
v
Q
if
p
v
Q
is
even
—
cf.
Remark
1.2.2,
(i).
Then,
roughly
speaking,
in
the
notation
of
Proposition
3.9,
(i),
the
mono-analytic
integral
structures
of
Proposition
3.2,
(ii),
in
n
∼
Q
α
I
(
F
v
Q
)
→
v
Q
k
i
;
∼
I
Q
(
A
F
v
Q
)
→
i=1
I
Q
(
α
F
v
Q
)
α∈A
are
given
by
n
∼
I(
F
v
Q
)
→
α
v
Q
I
i
;
∼
I(
A
F
v
Q
)
→
i=1
I(
α
F
v
Q
)
α∈A
while
the
local
holomorphic
integral
structures
O
α
F
v
Q
⊆
I
Q
(
α
F
v
Q
);
O
A
F
v
Q
⊆
I
Q
(
A
F
v
Q
)
of
Proposition
3.9,
(i),
in
the
ind-topological
rings
I
Q
(
α
F
v
Q
),
I
Q
(
A
F
v
Q
)
—
both
of
which
are
direct
sums
of
finite
extensions
of
Q
p
v
Q
—
are
given
by
the
subrings
of
integers
in
I
Q
(
α
F
v
Q
),
I
Q
(
A
F
v
Q
).
Thus,
by
applying
the
formula
of
the
final
display
of
[AbsTopIII],
Proposition
5.8,
(iii),
for
the
log-volume
of
I
i
,
[one
verifies
easily
that]
one
may
compute
the
log-volumes
α
μ
log
α,v
Q
(I(
F
v
Q
)),
A
μ
log
A,v
Q
(I(
F
v
Q
))
entirely
in
terms
of
the
given
initial
Θ-data.
We
leave
the
routine
details
to
the
reader.
(ii)
Suppose
that
v
Q
∈
V
arc
Q
.
Write
{v
1
,
.
.
.
,
v
n
v
}
for
the
[distinct!]
elements
Q
def
def
of
V
that
lie
over
v
Q
.
For
each
i
=
1,
.
.
.
,
n
v
Q
,
set
k
i
=
K
v
i
;
write
O
k
i
=
{λ
∈
k
i
|
|λ|
≤
1}
⊆
k
i
for
the
“set
of
integers”
of
k
i
,
def
I
i
=
π
·
O
k
i
⊆
k
i
—
cf.
Remark
1.2.2,
(ii).
Then,
roughly
speaking,
in
the
notation
of
Proposition
3.9,
(i),
the
discrepancy
between
the
mono-analytic
integral
structures
of
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
∼
Proposition
3.2,
(ii),
determined
by
the
I(
†
F
v
i
)
→
I
i
holomorphic
integral
structures
119
⊆
k
i
and
the
local
n
∼
O
α
F
v
Q
⊆
I
Q
(
α
F
v
Q
)
→
v
Q
k
i
i=1
∼
O
A
F
v
Q
⊆
I
Q
(
A
F
v
Q
)
→
I
Q
(
α
F
v
Q
)
α∈A
of
Proposition
3.9,
(i),
in
the
topological
rings
I
Q
(
α
F
v
Q
),
I
Q
(
A
F
v
Q
)
—
both
of
which
are
direct
sums
of
complex
archimedean
fields
—
determined
by
taking
the
product
[relative
to
this
direct
sum
decomposition]
of
the
respective
“subsets
of
integers”
may
be
computed
entirely
in
terms
of
the
given
initial
Θ-data,
by
applying
the
following
two
[easily
verified]
observations:
(a)
Equip
C
with
its
standard
Hermitian
metric,
i.e.,
the
metric
determined
by
the
complex
norm.
This
metric
on
C
determines
a
tensor
product
metric
on
C
⊗
R
C,
as
well
as
a
direct
sum
metric
on
C
⊕
C.
Then,
relative
to
these
metrics,
any
isomorphism
of
topological
rings
[i.e.,
arising
from
the
Chinese
remainder
theorem]
∼
C
⊗
R
C
→
C
⊕
C
is
compatible
with
these
metrics,
up
to
a
factor
of
2,
i.e.,
the
metric
on
the
right-hand
side
corresponds
to
2
times
the
metric
on
the
left-hand
side.
(b)
Relative
to
the
notation
of
(a),
the
direct
sum
decomposition
C
⊕
C,
together
with
its
Hermitian
metric,
is
preserved,
relative
to
the
displayed
isomorphism
of
(a),
by
the
operation
of
conjugation
on
either
of
the
two
copies
of
“C”
that
appear
√
in
C
⊗
R
C,
as
well
as
by
the
operations
of
multiplying
by
±1
or
±
−1
via
either
of
the
two
copies
of
“C”
that
appear
in
C
⊗
R
C.
We
leave
the
routine
details
to
the
reader.
(iii)
The
computation
of
the
discrepancy
between
local
holomorphic
and
mono-
analytic
integral
structures
will
be
discussed
in
more
detail
in
[IUTchIV],
§1.
Remark
3.9.2.
In
the
situation
of
Proposition
3.9,
(iii),
one
may
construct
[“mono-analytic”]
algorithms
for
recovering
the
subquotient
of
the
perfection
†
of
(
†
M
mod
)
α
=
(
M
MOD
)
α
associated
to
w
∈
V
[cf.
Remark
3.6.1],
together
with
the
submonoid
of
“nonnegative
elements”
of
such
a
subquotient,
by
considering
the
†
effect
of
multiplication
by
elements
of
(
†
M
mod
)
α
=
(
M
MOD
)
α
on
the
log-volumes
∼
defined
on
the
various
I
Q
(
A,α
F
v
)
→
I
Q
(
A,α
D
v
)
[cf.
Proposition
3.9,
(ii)].
Remark
3.9.3.
With
regard
to
the
procession-normalizations
discussed
in
Proposition
3.9,
(i),
(ii),
the
reader
might
wonder
the
following:
Is
it
possible
to
work
with
120
SHINICHI
MOCHIZUKI
more
general
weighted
averages,
i.e.,
as
opposed
to
just
averages,
in
the
usual
sense,
over
the
capsules
that
appear
in
the
procession?
The
answer
to
this
question
is
“no”.
Indeed,
in
the
situation
of
Proposition
3.4,
(ii),
for
j
∈
{1,
.
.
.
,
l
},
the
packet-normalized
log-volume
corresponding
to
the
capsule
with
index
set
S
±
j+1
may
be
thought
of
as
a
log-volume
that
arises
from
“any
one
of
the
log-shells
whose
label
∈
{0,
1,
.
.
.
,
j}”.
In
particular,
if
j
,
j
1
,
j
2
∈
{1,
.
.
.
,
l
},
and
j
≤
j
1
,
j
2
,
then
log-volumes
corresponding
to
the
same
log-shell
labeled
j
might
give
rise
to
packet-normalized
log-volumes
corresponding
to
either
±
of
[the
capsules
with
index
sets]
S
±
j
1
+1
,
S
j
2
+1
.
That
is
to
say,
in
order
for
the
resulting
notion
of
a
procession-normalized
log-volume
to
be
compatible
with
the
appearance
of
the
component
labeled
j
in
various
distinct
capsules
of
the
procession
—
i.e.,
compatible
with
the
various
inclusion
morphisms
of
the
procession!
—
one
has
no
choice
but
to
assign
the
same
weights
to
[the
capsules
with
index
sets]
±
S
±
j
1
+1
,
S
j
2
+1
.
Remark
3.9.4.
The
log-link
compatibility
of
log-volumes
discussed
in
Propo-
sition
3.9,
(iv),
may
be
formulated
somewhat
more
explicitly
by
applying
various
elementary
observations,
as
follows.
(i)
Let
(M,
μ
M
)
be
a
measure
space
[i.e.,
in
the
sense
of
the
discussion
of
Remark
3.1.1,
(iii)].
We
shall
say
that
a
subset
S
⊆
M
is
pre-ample
if
S
is
a
relatively
compact
Borel
set,
that
a
pre-ample
subset
S
⊆
M
is
ample
if
μ
M
(S)
>
0,
and
that
(M,
μ
M
)
is
ample
if
there
exists
an
ample
subset
of
M
.
In
the
following,
for
the
sake
of
simplicity,
we
assume
that
(M,
μ
M
)
is
ample.
Also,
to
simplify
the
notation,
we
shall
often
denote
the
dependence
of
objects
constructed
from
the
pair
(M,
μ
M
)
by
means
of
the
notation
“(M
)”
[i.e.,
as
opposed
to
“(M,
μ
M
)”].
Write
Sub(M
)
for
the
set
of
pre-ample
subsets
of
M
and
Fn(M
)
for
the
set
of
Borel
measurable
functions
f
:
M
→
R
≥0
such
that
the
image
f
(M
)
⊆
R
≥0
of
f
is
a
finite
set,
and,
moreover,
M
⊇
f
−1
(R
>0
)
∈
Sub(M
).
Observe
that
Fn(M
)
is
equipped
with
a
natural
monoid
structure
[induced
by
the
natural
monoid
structure
on
R
≥0
],
as
well
as
a
natural
action
by
R
>0
[induced
by
the
natural
action
by
multiplication
of
R
>0
on
R
≥0
].
By
assigning
to
an
element
S
∈
Sub(M
)
the
characteristic
function
χ
S
:
M
→
R
≥0
[i.e.,
which
is
=
1
on
S
and
=
0
on
M
\S],
we
shall
regard
Sub(M
)
as
a
subset
of
Fn(M
).
Note
that
integration
over
M
,
relative
to
the
measure
μ
M
,
determines
an
R
>0
-equivariant
surjection
:
Fn(M
)
R
≥0
M
whose
restriction
to
Sub(M
)
maps
Sub(M
)
S
→
μ
M
(S)
∈
R
≥0
.
In
particular,
if
we
write
FnRss
M
:
Fn(M
)
Rss(M
)
for
the
natural
map
to
the
quotient
set
of
Fn(M
)
[i.e.,
the
set
of
equivalence
classes
of
elements
of
Fn(M
)]
determined
by
M
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
121
[so
Rss(M
)
also
admits
a
natural
monoid
structure,
as
well
as
a
natural
action
by
R
>0
],
then
we
obtain
a
natural
R
>0
-equivariant
isomorphism
of
monoids
Rss
∼
:
Rss(M
)
→
R
≥0
M
such
that
M
=
Rss
◦FnRss
M
.
Here,
M
we
wish
to
think
of
integration
M
—
and
hence
of
the
quotient
FnRss
M
:
Fn(M
)
Rss(M
)
—
as
a
sort
of
“realified
semi-simplification”
of
(M,
μ
M
)
[i.e.,
roughly
in
the
spirit
of
the
Grothendieck
group
associated
to
an
additive
cat-
egory],
that
is
to
say,
a
quotient
in
the
category
of
commutative
monoids
with
R
>0
-action,
whose
restriction
to
Sub(M
)
⊆
Fn(M
)
·
identifies
S
1
,
S
2
∈
Sub(M
)
such
that
μ
M
(S
1
)
=
μ
M
(S
2
)
[such
as,
for
instance,
additive
translates
of
an
element
S
∈
Sub(M
)
relative
to
an
additive
structure
on
M
with
respect
to
which
μ
M
is
invariant];
·
maps
S
1
∪
S
2
∈
Sub(M
)
to
the
sum
[relative
to
the
monoid
structure
on
the
quotient]
of
the
images
of
S
1
,
S
2
∈
Sub(M
)
whenever
S
1
,
S
2
∈
Sub(M
)
are
disjoint
[i.e.,
as
subsets
of
M
].
We
shall
refer
to
a
subset
E
⊆
Fn(M
)
as
ample
if
(R
≥0
⊇)
R
>0
∩
M
(E)
=
∅.
Thus,
if,
for
instance,
S
∈
Sub(M
)
is
ample
and
compact,
then
the
pair
(S,
μ
M
|
S
)
obtained
by
restricting
μ
M
to
S
is
an
ample
measure
space
that
determines
compatible
natural
inclusions
Sub(S)
→
Sub(M
),
Fn(S)
→
Fn(M
)
[the
latter
of
which
is
defined
by
extension
by
zero]
—
which
we
shall
use
to
regard
Sub(S),
Fn(S)
as
subsets
of
Sub(M
),
Fn(M
),
respectively
—
such
that
the
subsets
Sub(S),
Fn(S)
⊆
Fn(M
)
are
ample.
If
E
⊆
Fn(M
)
is
ample,
then
the
image
of
E
in
Rss(M
)
determines
a
natural
subset
Rss(E)
⊆
Rss(M
),
whose
R
≥0
-orbit
R
≥0
·
Rss(E)
is
equal
to
Rss(M
).
In
particular,
if
S
∈
Sub(M
)
is
ample
and
compact,
then
we
obtain
natural
R
>0
-
equivariant
isomorphisms
of
monoids
∼
∼
Rss(S)
→
R
≥0
·
Rss(Fn(S))
→
Rss(M
)
—
where,
the
notation
“R
≥0
·
Rss(Fn(S))”
is
intended
relative
to
the
interpre-
tation
of
Fn(S)
as
a
subset
of
Fn(M
)
—
such
that
the
composite
isomorphism
∼
∼
Rss
Rss(S)
→
Rss(M
)
is
compatible
with
the
isomorphisms
S
:
Rss(S)
→
R
≥0
,
∼
Rss
:
Rss(M
)
→
R
≥0
.
Finally,
we
observe
that
if
(M
1
,
μ
M
1
)
and
(M
2
,
μ
M
2
)
are
M
ample
measure
spaces,
then
the
product
measure
space
(M
1
×
M
2
,
μ
M
1
×M
2
)
is
also
an
ample
measure
space;
moreover,
there
is
a
natural
map
Sub(M
1
)
×
Sub(M
2
)
→
Sub(M
1
×
M
2
)
that
maps
(S
1
,
S
2
)
→
S
1
×
S
2
and
induces
a
natural
R
>0
-equivariant
isomorphism
of
monoids
∼
Rss(M
1
)
⊗
Rss(M
2
)
→
Rss(M
1
×
M
2
)
122
SHINICHI
MOCHIZUKI
that
is
compatible
with
the
isomorphisms
∼
Rss
Rss
⊗
M
2
:
Rss(M
1
)
⊗
Rss(M
2
)
→
R
≥0
,
M
1
∼
Rss
:
Rss(M
1
×
M
2
)
→
R
≥0
.
[Here,
we
observe
that
there
is
a
natural
notion
of
M
1
×M
2
“tensor
product
of
monoids
isomorphic
to
R
≥0
”
since
such
a
monoid
may
be
thought
of,
by
passing
to
the
groupification
of
such
a
monoid,
as
a
one-dimensional
R-vector
space
equipped
with
a
subset
[which
forms
a
R
>0
-torsor]
of
“positive
elements”.]
(ii)
One
very
rough
approach
to
understanding
the
log-link
compatibility
of
log-
volumes
is
the
following.
Suppose
that
instead
of
knowing
this
property,
one
only
knows
that
each
application
of
the
log-link
has
the
effect
of
dilating
volumes
by
a
factor
λ
∈
R
>0
\
{1}.
[Here,
relative
to
the
notation
of
(i),
we
observe
that
this
sort
of
situation
in
which
volumes
are
dilated
in
a
nontrivial
fashion
may
be
seen
in
the
following
example:
def
Suppose
that
M
=
Q
p
,
for
some
prime
number
p,
equipped
with
the
[additive]
Haar
measure
μ
Q
p
normalized
so
that
Z
p
⊆
Q
p
has
measure
1,
so
(Q
p
,
μ
Q
p
)
is
an
ample
measure
space
in
the
sense
of
(i).
Then
∼
multiplication
by
p
induces
a
bijection
α
p
:
Q
p
→
Q
p
.
Moreover,
α
p
∼
induces
compatible
bijections
Sub(α
p
)
:
Sub(Q
p
)
→
Sub(Q
p
),
Fn(α
p
)
:
∼
∼
Fn(Q
p
)
→
Fn(Q
p
),
Rss(α
p
)
:
Rss(Q
p
)
→
Rss(Q
p
).
On
the
other
hand,
[unlike
the
situation
discussed
in
(i)
concerning
the
“composite
isomor-
∼
phism
Rss(S)
→
Rss(M
)”!]
in
the
present
context,
Rss(α
p
)
is
not
com-
∼
Rss
patible
with
the
isomorphisms
Q
p
:
Rss(Q
p
)
→
R
≥0
in
the
domain
and
codomain
of
Rss(α
p
),
i.e.,
it
is
only
compatible
up
to
a
factor
p
−1
(
=
1)!]
Then
in
order
to
compute
log-volumes
in
a
fashion
that
is
consistent
with
the
various
arrows
[i.e.,
both
Kummer
isomorphisms
and
log-links!]
of
the
“systems”
consti-
tuted
by
the
log-Kummer
correspondences
discussed
in
Proposition
3.5,
(ii),
it
would
be
necessary
to
regard
the
various
“log-volumes”
computed
as
only
giving
rise
to
well-defined
elements
[not
∈
R,
but
rather]
∈
R/Z
·
log(λ)
(
∼
=
S
1
)
—
a
situation
which
is
not
acceptable,
relative
to
the
goal
of
obtaining
log-volume
estimates
[i.e.,
as
in
Corollary
3.12
below]
for
the
various
objects
for
which
log-shells
serve
as
“multiradial
containers”
[cf.
the
discussion
of
Remark
1.5.2;
the
content
of
Theorem
3.11
below].
(iii)
In
the
following
discussion,
we
use
the
notation
of
Remark
1.2.2,
(i).
Thus,
we
regard
k
as
being
equipped
with
the
[additive]
Haar
measure
μ
k
normalized
so
that
μ
k
(O
k
)
=
1
[cf.
[AbsTopIII],
Proposition
5.7,
(i)].
Then
(k,
μ
k
)
is
an
ample
measure
space
in
the
sense
of
(i);
O
k
×
⊆
k
is
an
ample
subset;
for
any
compact
ample
subset
S
⊆
O
k
×
on
which
log
k
:
O
k
×
→
k
is
injective,
we
have
μ
k
(S)
=
μ
k
(log
k
(S))
[cf.
[AbsTopIII],
Proposition
5.7,
(i),
(c)].
In
particular,
by
applying
the
formalism
of
realified
semi-simplifications
introduced
in
(i),
we
conclude
that
the
diagram
k
⊇
O
k
×
log
k
−→
∪
S
log
k
(O
k
×
)
∪
∼
→
log
k
(S)
⊆
k
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
123
induces
a
commutative
diagram
∼
Rss(O
k
×
)
⏐
Rss
⏐
O
k
×
←
Rss(S)
→
⏐
⏐
Rss
S
=
R
≥0
=
R
≥0
Rss(k)
←
⏐
⏐
Rss
k
R
≥0
∼
∼
Rss(log
k
(S))
⏐
⏐
Rss
log
k
(S)
=
R
≥0
∼
→
∼
Rss(log
k
(O
k
×
))
→
⏐
Rss
⏐
log
k
(O
k
×
)
Rss(k)
⏐
⏐
Rss
k
=
R
≥0
=
R
≥0
—
where
the
vertical
arrows
are
R
>0
-equivariant
isomorphisms
of
monoids,
and
the
∼
composite
[R
>0
-equivariant]
isomorphism
[of
monoids]
Rss(O
k
×
)
→
Rss(log
k
(O
k
×
))
is
easily
verified
to
be
independent
of
the
choice
of
the
compact
ample
subset
S
⊆
O
k
×
.
[Also,
we
observe
that
it
is
easily
verified
that
there
exist
compact
ample
subsets
S
⊆
O
k
×
for
which
the
induced
map
S
log
k
(S)
is
injective.]
One
may
then
compose
this
diagram
with
the
bijection
∼
log
:
R
≥0
→
R
∪
{−∞}
determined
by
the
natural
logarithm
and
then
multiply
by
a
suitable
normalization
factor
∈
R
>0
to
conclude
that
the
diagram
⊇
k
O
k
×
log
k
−→
log
k
(O
k
×
)
⊆
k
induces
R
>0
-equivariant
isomorphisms
of
monoids
on
the
respective
realified
semi-simplifications
“Rss(−)”,
all
of
which
are
compatible
with
the
log-volume
maps
on
each
of
the
“Rss(−)’s”,
i.e.,
which
restrict
to
the
“usual
log-volume
maps”
on
the
respective
“Sub(−)’s”,
relative
to
the
natural
maps
“Sub(−)
→
Rss(−)”.
This
is
one
way
to
formulate
the
log-link
compatibility
of
log-volumes
discussed
in
Proposition
3.9,
(iv),
in
the
case
of
v
∈
V
non
.
Finally,
we
observe
that
this
log-link
compatibility
with
log-volumes
is
itself
compatible
with
passing
to
finite
extensions
of
k
[or,
more
generally,
Q
p
v
],
as
follows.
Let
k
1
⊆
k
2
be
finite
field
extensions
of
Q
p
v
.
We
shall
use
analogous
notation
for
objects
associated
to
k
1
and
k
2
to
the
notation
that
was
used
above
for
objects
associated
to
k.
Then
observe
that
since
O
k
2
is
a
finite
free
O
k
1
-module
of
rank
[k
2
:
k
1
],
it
follows
that
the
[additive]
compact
topological
group
O
k
2
is
isomorphic
to
the
product
of
[k
2
:
k
1
]
copies
of
the
[additive]
compact
topological
group
O
k
1
.
In
particular,
since
the
Haar
measure
of
a
compact
topological
group
is
invariant
with
respect
to
arbitrary
automorphisms
of
the
topological
group,
we
thus
conclude
[cf.
the
discussion
of
product
measure
spaces
in
(i)]
that
the
inclusion
of
topological
fields
k
1
→
k
2
induces
natural
R
>0
-
equivariant
isomorphisms
of
monoids
∼
∼
∼
Rss(k
1
)
⊗[k
2
:k
1
]
←
Rss(O
k
1
)
⊗[k
2
:k
1
]
→
Rss(O
k
2
)
→
Rss(k
2
)
∼
such
that
the
composite
isomorphism
Rss(k
1
)
⊗[k
2
:k
1
]
→
Rss(k
2
)
is
compatible
with
the
R
>0
-equivariant
isomorphisms
of
monoids
Rss
k
1
⊗[k
2
:k
1
]
∼
:
Rss(k
1
)
⊗[k
2
:k
1
]
→
R
≥0
124
and
SHINICHI
MOCHIZUKI
∼
Rss
:
Rss(k
2
)
→
R
≥0
.
k
2
def
(iv)
In
the
notation
of
(iii),
suppose
that
v
∈
V
bad
;
write
q
=
q
.
Thus,
we
v
have
a
submonoid
O
k
×
×
q
N
⊆
k
of
the
underlying
multiplicative
monoid
of
k.
Then
the
various
arrows
of
the
log-
Kummer
correspondence
discussed
in
Proposition
3.5,
(ii),
may
be
thought
of,
from
the
point
of
view
of
a
vertically
coric
étale
holomorphic
copy
of
“k”
[i.e.,
a
copy
labeled
“n,
◦”,
as
in
Proposition
3.5,
(i)],
as
corresponding
to
the
operations
k
O
k
×
×
q
N
(⊆
k)
O
k
×
(⊆
k)
log
k
(O
k
×
)
(⊆
k)
k
—
i.e.,
of
passing
first
from
k
to
the
multiplicative
submonoid
O
k
×
×
q
N
,
then
to
the
multiplicative
submonoid
O
k
×
,
then
applying
log
k
to
obtain
an
additive
submonoid
of
k,
and
finally
passing
from
this
submonoid
back
to
k
itself.
Then
the
log-
volume
compatibility
discussed
in
(iii)
may
be
understood,
in
the
context
of
the
log-Kummer
correspondence,
as
the
statement
that
the
operations
of
the
above
display
induce
R
>0
-equivariant
isomorphisms
of
monoids
∼
∼
∼
Rss(k)
→
Rss(O
k
×
)
→
Rss(log
k
(O
k
×
))
→
Rss(k)
that
are
compatible
with
the
respective
[normalized]
log-volume
maps
to
R
∪
{−∞}
[cf.
the
discussion
of
(iii)],
in
such
a
way
as
to
avoid
any
interference,
up
to
multiplication
by
roots
of
unity,
with
the
submonoid
q
N
⊆
k,
which
induces,
by
applying
the
[normalized]
log-volume
to
the
image
of
O
k
⊆
k
via
multiplication
by
elements
of
this
submonoid,
an
embedding
∼
N
→
Rss(k)
→
R
∪
{−∞}
that
maps
N
1
→
−log(q)
∈
R
def
[where
we
write
log(q)
=
ord
v
(q
)
·
log(p
v
)
∈
R
—
cf.
the
notation
of
Remark
v
2.4.2,
(ii)].
A
similar
interpretation
of
log-volume
compatibility
in
the
context
of
the
log-Kummer
correspondence
may
be
given
in
the
case
of
v
∈
V
good
V
non
by
simply
omitting
the
portion
of
the
above
discussion
concerning
“q”.
(v)
In
the
notation
of
Remark
3.9.1,
(i),
we
observe
that
the
discussion
of
(iii),
(iv),
may
be
extended
to
topological
tensor
products
of
the
form
def
k
i
A
=
k
i
α
α∈A
—
where
i
α
∈
{1,
.
.
.
,
n
v
Q
},
for
each
α
∈
A,
and
we
regard
k
i
A
as
being
equipped
with
the
[additive]
Haar
measure
normalized
[cf.
Proposition
3.9,
(i)]
so
that
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
125
ring
of
integers
O
k
iA
⊆
k
i
A
[i.e.,
the
integral
structure
discussed
in
Proposition
3.1,
(ii)]
has
Haar
measure
=
1.
Indeed,
each
of
the
direct
summand
fields
of
k
i
A
[cf.
Proposition
3.1,
(i)]
may
be
taken
to
be
a
[finite
extension
of
a]
“k”
as
in
(iii),
(iv).
In
particular,
the
measure
space
k
i
A
may
be
regarded
as
a
product
measure
space
of
[finite
extensions
of]
“k”
as
in
(iii),
(iv),
so
one
may
extend
(iii),
(iv)
to
k
i
A
by
applying
(iii),
(iv)
to
each
factor
of
this
product
measure
space
[cf.
the
discussion
of
product
measure
spaces
in
(i)].
[We
leave
the
routine
details
to
the
reader.]
On
the
other
hand,
in
this
context,
it
is
also
of
interest
to
observe
that
it
follows
immediately
from
the
discussion
of
compatibility
with
finite
extensions
in
(iii),
together
with
the
discussion
of
product
measure
spaces
in
(i),
that,
for
each
α
∈
A,
the
natural
structure
of
k
i
A
as
a
k
i
α
-algebra
determines
natural
R
>0
-
equivariant
isomorphisms
of
monoids
∼
∼
Rss(Q
p
v
)
⊗d
A
→
Rss(k
i
α
)
⊗d
α
→
Rss(k
i
A
)
—
where
we
write
d
α
particular,
def
=
β∈A\{α}
[k
i
β
:
Q
p
v
],
d
A
def
=
d
α
·
[k
i
α
:
Q
p
v
].
In
the
log-link
compatibility
of
log-volumes
[as
discussed
above]
for
the
realified
semi-simplification
Rss(k
i
A
)
of
the
topological
tensor
product
k
i
A
may
be
understood,
for
any
α
∈
A,
as
the
[functorially
induced!]
d
α
-th
tensor
power
of
the
log-link
compatibility
of
log-volumes
for
the
realified
semi-simplification
Rss(k
i
α
)
of
k
i
α
or,
alternatively/equivalently,
as
the
[functorially
induced!]
d
A
-th
tensor
power
of
the
log-link
compatibility
of
log-volumes
for
the
realified
semi-simplification
Rss(Q
p
v
)
of
Q
p
v
—
where
we
note
that
the
latter
“alternative/equivalent”
approach
has
the
virtue
of
being
independent
of
the
choice
of
α
∈
A.
(vi)
In
the
following
discussion,
we
use
the
notation
of
Remark
1.2.2,
(ii).
We
regard
the
complex
archimedean
field
k
as
being
equipped
with
the
standard
Euclidean
metric
[cf.
the
discussion
of
“metrics”
in
Remark
1.2.1,
(ii)],
with
respect
to
which
O
k
×
⊆
k
has
length
2π.
This
metric
on
k
thus
determines
measures
μ
|k|
on
|k|
=
k/O
k
×
and
μ
O
×
on
O
k
×
⊆
k
[cf.
the
situation
discussed
in
[AbsTopIII],
def
k
Proposition
5.7,
(ii)]
such
that
(|k|,
μ
|k|
)
and
(O
k
×
,
μ
O
×
)
are
ample
measure
spaces
k
in
the
sense
of
(i).
Moreover,
by
(a)
thinking
of
O
k
×
as
a
union
of
closed
arcs
[i.e.,
whose
interiors
are
disjoint]
of
measure
μ
O
×
(−)
<
,
for
some
positive
real
number
,
k
(b)
considering
additive
translates
of
such
closed
arcs
that
map
one
of
the
endpoints
of
the
arc
to
0
∈
k,
126
SHINICHI
MOCHIZUKI
(c)
projecting
such
additive
translates
via
the
natural
surjection
k
|k|,
and
(d)
passing
to
the
limit
→
0,
one
verifies
immediately
that
we
obtain,
by
applying
the
formalism
of
realified
semi-simplifications
introduced
in
(i),
a
natural
R
>0
-equivariant
isomorphism
of
∼
monoids
ρ
k
:
Rss(O
k
×
)
→
Rss(|k|),
together
with
a
commutative
diagram
Rss(|k|)
⏐
⏐
Rss
|k|
←
∼
Rss(O
k
×
)
⏐
Rss
⏐
O
k
×
←
∼
R
≥0
=
R
≥0
=
∼
Rss(|k|)
⏐
⏐
Rss
|k|
=
R
≥0
Rss(log
k
(O
k
×
))
→
⏐
Rss
⏐
log
k
(O
k
×
)
R
≥0
—
where
the
vertical
arrows
are
R
>0
-equivariant
isomorphisms
of
monoids;
the
first
∼
def
×
“
←
”
is
ρ
k
;
we
regard
log
k
(O
k
×
)
=
exp
−1
k
(O
k
)
as
being
equipped
with
the
measure
μ
log
k
(O
×
)
[such
that
(log
k
(O
k
×
),
μ
log
k
(O
×
)
)
is
an
ample
measure
space]
obtained
by
k
k
∼
pulling
back
μ
|k|
via
the
homeomorphism
log
k
(O
k
×
)
→
|k|
induced
by
restricting
∼
the
natural
surjection
k
|k|
to
log
k
(O
k
×
)
⊆
k;
the
second
“
←
”
is
the
natural
R
>0
-equivariant
isomorphism
of
monoids
naturally
induced
[i.e.,
by
considering
ample
S
⊆
log
k
(O
k
×
)
that
map
bijectively
to
exp
k
(S)
⊆
O
k
×
—
cf.
[AbsTopIII],
Proposition
5.7,
(ii),
(c)]
by
the
universal
covering
map
exp
k
|
log
k
(O
×
)
:
log
k
(O
k
×
)
→
k
∼
O
k
×
;
the
“
→
”
is
the
natural
R
>0
-equivariant
isomorphism
of
monoids
induced
by
∼
the
homeomorphism
log
k
(O
k
×
)
→
|k|.
One
may
then
compose
this
diagram
with
the
bijection
∼
log
:
R
≥0
→
R
∪
{−∞}
determined
by
the
natural
logarithm
and
then
multiply
by
a
suitable
normalization
factor
∈
R
>0
to
conclude
that
the
diagram
|k|
k
⊇
O
k
×
exp
k
←−
log
k
(O
k
×
)
⊆
k
|k|
induces
R
>0
-equivariant
isomorphisms
of
monoids
on
the
respective
realified
semi-simplifications
“Rss(−)”
of
|k|,
O
k
×
,
log
k
(O
k
×
),
and
|k|;
each
of
these
isomorphisms
is
compatible
with
the
log-volume
map
on
“Rss(−)”,
i.e.,
which
restricts
to
the
“usual
radial/angular
log-volume
map”
on
“Sub(−)”
[that
is
to
say,
the
map
uniquely
determined
by
the
radial/angular
log-volume
map
of
[AbsTopIII],
Proposition
5.7,
(ii),
(a)]
relative
to
the
natural
map
“Sub(−)
→
Rss(−)”.
This
is
one
way
to
formulate
the
log-link
compatibility
of
log-volumes
discussed
in
Proposition
3.9,
(iv),
in
the
case
of
v
∈
V
arc
.
One
verifies
immediately
that
one
also
has
analogues
for
v
∈
V
arc
of
(iv),
(v).
[We
leave
the
routine
details
to
the
reader.]
Remark
3.9.5.
In
situations
that
involve
consideration
of
various
sorts
of
regions
[cf.
the
discussion
of
Remarks
3.1.1,
(iii),
(iv);
3.9.4]
to
which
the
log-volume
may
be
applied,
it
is
often
of
use
to
consider
the
notion
of
the
holomorphic
hull
of
a
region.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
127
(i)
Suppose
that
we
are
in
the
situation
of
Proposition
3.9,
(i).
Let
α
U
⊆
I
Q
(
α
F
v
Q
)
(respectively,
A
U
⊆
I
Q
(
A
F
v
Q
);
A,α
U
⊆
I
Q
(
A,α
F
v
))
be
a
subset
that
contains
a
relatively
compact
subset
whose
log-volume
[cf.
the
discussion
of
Remark
3.1.1,
(iii),
(iv),
as
well
as
Remark
3.9.7,
(ii),
below]
is
finite
[i.e.,
>
−∞].
If
α
U
(respectively,
A
U;
A,α
U)
is
relatively
compact,
then
we
define
the
holomorphic
hull
of
α
U
(respectively,
A
U
;
A,α
U)
to
be
the
smallest
subset
of
the
form
α
def
H
=
λ
·
O
α
F
v
Q
def
(respectively,
A
H
=
λ
·
O
A
F
v
Q
;
A,α
def
H
=
λ
·
O
A,α
F
v
)
—
where,
relative
to
the
direct
sum
decomposition
of
I
Q
((−))
as
a
direct
sum
of
fields
[cf.
the
discussion
of
Proposition
3.9,
(i)],
λ
∈
I
Q
((−))
is
an
element
such
that
each
component
of
λ
[i.e.,
relative
to
this
direct
sum
decomposition]
is
nonzero
—
that
contains
α
U
(respectively,
A
U
;
A,α
U
).
If
α
U
(respectively,
A
U
;
A,α
U
)
is
not
relatively
compact,
then
we
define
the
holomorphic
hull
of
α
U
(respectively,
A
U
;
A,α
U
)
to
be
I
Q
(
α
F
v
Q
)
(respectively,
I
Q
(
A
F
v
Q
);
I
Q
(
A,α
F
v
)).
One
verifies
immedi-
ately
that
the
holomorphic
hull
is
well-defined
[under
the
conditions
stated].
(ii)
In
the
remainder
of
the
discussion
of
the
present
Remark
3.9.5,
for
the
sake
of
simplicity,
we
shall
refer
to
“holomorphic
hulls”
as
“hulls”.
Write
def
P
=
{P
⊆
I
Q
((−))
|
P
is
a
direct
product
region
[cf.
Remark
3.1.1,
(iii)]};
def
H
=
{H
I
Q
((−))
|
H
is
a
hull
[cf.
(i)]}
—
where
the
argument
“(−)”
is
“
α
F
v
Q
”,
“
A
F
v
Q
”,
or
“
A,α
F
v
”
[cf.
(i)],
and
we
observe
that
H
⊆
P.
Then
it
is
essentially
a
tautology
that
the
operation
of
forming
the
hull
discussed
in
(i)
U
→
H
—
where
“”
is
“α”,
“A”,
or
“A,
α”
—
determines
a
map
φ
:
P
→
H
that
may
be
characterized
uniquely
by
the
following
properties
(P1)
φ(H)
=
H,
for
any
H
∈
H;
(P2)
P
⊆
φ(P
),
for
any
P
∈
P;
(P3)
φ(P
1
)
⊆
φ(P
2
),
for
any
P
1
,
P
2
∈
P
such
that
P
1
⊆
P
2
.
Indeed,
since,
as
is
easily
verified,
any
intersection
of
elements
of
H
which
is
of
finite
log-volume
necessarily
determines
an
element
of
H,
it
follows
formally
from
(P1),
(P2),
(P3)
that
φ(P
)
=
H
HH⊇P
for
any
P
∈
P.
Put
another
way,
this
map
φ
may
be
thought
of
as
a
sort
of
adjoint,
or
push
forward
in
the
opposite
direction,
of
the
inclusion
H
⊆
P.
Alternatively,
φ
may
be
thought
of
as
a
sort
of
canonical
splitting
of
the
inclusion
H
⊆
P,
or,
in
the
spirit
of
the
discussion
of
Remark
3.9.4,
as
a
sort
of
integration
operation.
The
compatibility
[cf.
(P2),
(P3)]
of
φ
with
the
pre-order
structure
on
P
determined
128
SHINICHI
MOCHIZUKI
by
inclusion
of
direct
product
regions
will
play
an
important
role
in
the
context
of
various
log-volume
estimates
of
regions.
(iii)
Now
we
consider
the
various
log-volumes
μ
log
(−)
[where
the
argument
“(−)”
is
“α,
v
Q
”,
“A,
v
Q
”,
or
“A,
α,
v”
—
cf.
(ii)]
of
Proposition
3.9,
(i)
[cf.
also
Remark
3.1.1,
(iii)],
in
the
context
of
the
discussion
of
(ii).
In
the
following,
for
the
sake
of
log
simplicity,
we
shall
denote
“μ
log
.
For
P
∈
P,
write
(−)
”
by
μ
def
Φ(P
)
=
{H
∈
H
|
φ(P
)
⊇
H,
(μ
log
(φ(P
))
≥)
μ
log
(H)
≥
μ
log
(P
)}
⊆
H;
def
Ξ(P
)
=
{H
∈
H
|
φ(P
)
⊇
H,
(μ
log
(φ(P
))
≥)
μ
log
(H)
=
μ
log
(P
)}
⊆
Φ(P
);
def
def
H
Φ(P
)
=
H
⊆
φ(P
);
H
Ξ(P
)
=
H
⊆
H
Φ(P
)
⊆
φ(P
).
H∈Φ(P
)
H∈Ξ(P
)
Thus,
one
may
think
of
elements
∈
Φ(P
)
or
∈
Ξ(P
)
as
“log-volume
approximations”
of
P
by
means
of
hulls
∈
H.
If
one
thinks
of
distinct
elements
∈
Φ(P
)
or
∈
Ξ(P
)
—
i.e.,
of
the
issue
of
con-
structing
a
“log-volume
hull-approximant”
of
P
—
as
a
sort
of
indeterminacy
[i.e.,
in
the
assignment
to
P
of
a
specific
element
∈
H!],
then
this
indeterminacy
is
compact,
i.e.,
in
the
sense
that
all
possible
choices
of
an
element
∈
Φ(P
)
or
∈
Ξ(P
)
are
contained
in
the
compact
set
φ(P
)
∈
H.
Indeed,
developing
the
theory
in
such
a
way
that
all
the
indeterminacies
that
occur
in
the
theory
are
compact
is
in
some
sense
one
important
theme
in
the
present
series
of
papers.
Note
that
this
compactness
would
not
be
valid
if,
in
the
definition
of
Φ(−)
or
Ξ(−),
one
omits
the
condition
“H
⊆
φ(P
)”.
(iv)
In
the
context
of
(iii),
we
observe
that
φ(P
)
∈
Φ(P
),
so
φ(P
)
=
H
Φ(P
)
,
but
the
issue
of
whether
or
not
Ξ(P
)
=
∅
is
not
so
immediate.
Indeed:
(Ξ1)
If
either
of
the
following
conditions
is
satisfied,
then
it
is
easily
verified
that
Ξ(P
)
=
∅:
(Ξ
non
1)
if
we
write
K
cl
for
the
Galois
closure
of
K
over
Q,
then
the
residue
field
extension
degree
of
each
valuation
∈
V(K
cl
)
that
is
=
1,
and,
moreover,
μ
log
(P
)
=
μ
log
(Q),
for
divides
v
Q
∈
V
non
Q
some
Q
∈
P
which
is
a
Z
p
v
Q
-submodule
of
I
Q
((−));
(Ξ
arc
1)
v
Q
∈
V
arc
Q
.
(Ξ2)
If
one
allows
the
v
Q
∈
V
Q
in
the
present
discussion
to
vary,
and
one
con-
siders
global
situations
[i.e.,
which
necessarily
involve
the
unique
valuation
∈
V
arc
Q
!]
as
in
Proposition
3.9,
(iii),
then
it
is
easily
verified
that
the
global
analogue
of
“Ξ(P
)”
is
nonempty.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
129
On
the
other
hand,
in
general,
it
is
not
so
clear
whether
or
not
Ξ(P
)
=
∅.
In
this
context,
it
is
also
of
interest
to
observe
that
if
P
∈
H,
then
{φ(P
)}
=
Φ(P
)
=
Ξ(P
),
so
P
=
φ(P
)
=
H
Φ(P
)
=
H
Ξ(P
)
,
but
in
general,
even
in
the
situation
of
(Ξ1),
the
inclusion
P
⊆
φ(P
)
=
H
Φ(P
)
,
as
well
as
the
induced
inequality
of
log-volumes
μ
log
(P
)
≤
μ
log
(H
Φ(P
)
)
=
μ
log
(φ(P
)),
is
strict.
Indeed,
for
instance,
(Ξ3)
in
the
situation
of
(Ξ1),
if
(V
mod
)
v
Q
[cf.
the
notational
conventions
of
Remark
3.1.1,
(iii)]
and
A
are
of
cardinality
≥
2,
then
one
verifies
easily
that
there
exist
P
∈
P
for
which
μ
log
(P
)
<
μ
log
(H
Ξ(P
)
)
(≤
μ
log
(H
Φ(P
)
)
=
μ
log
(φ(P
))).
This
sort
of
phenomenon
may
be
seen
in
the
following
example:
def
Let
p
be
a
prime
number.
Write
I
=
Q
p
×
Q
p
;
def
H
0
=
Z
p
×
Z
p
⊆
I;
def
H
1
=
(p
−1
·
Z
p
)
×
(p
·
Z
p
)
⊆
I;
def
P
=
H
0
∪
({p
−1
}
×
Z
p
)
⊆
I;
def
H
P
=
(p
−1
·
Z
p
)
×
Z
p
⊆
I
—
where
we
think
of
I
as
being
equipped
with
the
Haar
measure
μ
I
nor-
malized
so
that
μ
I
(H
0
)
=
1.
Thus,
p
corresponds
to
“p
v
Q
”
such
that
“(V
mod
)
v
Q
”
is
of
cardinality
2;
I
corresponds
to
“I
Q
(
α
F
v
Q
)”;
P
corre-
sponds
to
“P
”;
H
P
corresponds
to
“φ(P
)”;
H
0
and
H
1
correspond
to
elements
of
“Ξ(P
)”,
so
H
0
∪
H
1
corresponds
to
a
subset
of
“H
Ξ(P
)
”,
hence
also
a
subset
of
“H
Φ(P
)
=
φ(P
)”.
Then
μ
I
(P
)
=
μ
I
(H
0
)
=
1
<
2
−
p
−1
=
μ
I
(H
0
∪
H
1
)
—
i.e.,
the
inequality
of
[log-]volumes
in
question
is
strict.
In
fact,
by
considering
various
translates
of
H
0
,
H
1
by
automorphisms
of
the
Z
p
-
module
H
P
,
one
verifies
immediately
that
H
P
corresponds
not
only
to
“φ(P
)
=
H
Φ(P
)
”,
but
also
to
“H
Ξ(P
)
”.
That
is
to
say,
this
is
a
situa-
tion
in
which
one
has
“H
Ξ(P
)
=
H
Φ(P
)
=
φ(P
)”,
hence
also
“μ
log
(P
)
<
μ
log
(H
Ξ(P
)
)
=
μ
log
(H
Φ(P
)
)
=
μ
log
(φ(P
))”.
(v)
Let
E
be
a
set,
S
⊆
E
a
proper
subset
of
E
of
cardinality
≥
2
[so
S
=
∅
=
E
\
S].
Write
def
{S}
(E
)
E
S
=
(E
\
S)
[i.e.,
“E
upper
S”]
for
the
set-theoretic
quotient
of
E
by
S,
i.e.,
the
quotient
of
E
obtained
by
identifying
the
elements
of
S
and
leaving
E
\
S
unaffected.
Write
def
S
=
{S}
⊆
E
S.
Then
observe
that
any
set-theoretic
map
(E
⊇)
S
1
→
S
2
(⊆
E)
between
nonempty
subsets
S
1
,
S
2
⊆
S
(⊆
E)
induces,
upon
passing
to
the
quotient
E
E
S,
the
identity
map
(E
S
⊇)
S
→
S
(⊆
E
S)
130
SHINICHI
MOCHIZUKI
between
the
images
[i.e.,
both
of
which
are
equal
to
S
!]
of
S
1
,
S
2
in
E
S,
hence
lies
over
the
identity
map
E
S
→
E
S
on
E
S.
Moreover,
this
map
may
be
“extended”
to
the
case
where
S
i
[for
i
∈
{1,
2}]
is
empty
if
this
S
i
is
treated
as
a
“formal
intersection”
[cf.
our
hypothesis
that
the
cardinality
of
S
is
≥
2]
—
i.e.,
a
“category-theoretic
formal
fiber
product,
or
inverse
system,
over
E”
—
of
some
collection
of
nonempty
subsets
of
S.
That
is
to
say,
such
an
inverse
system
induces,
upon
passing
to
the
quotient
E
E
S,
a
system
that
consists
of
identity
maps
between
copies
of
S
.
In
particular,
if
one
thinks
in
terms
of
such
formal
inverse
systems,
then
“formal
empty
sets”
⊆
S
(⊆
E)
also
map
to
S
⊆
E
S.
Finally,
we
observe
that
the
above
discussion
may
be
thought
of
as
an
abstract
set-theoretic
formalization
of
the
notions
of
upper
semi-
commutativity/semi-compatibility,
as
discussed
in
Remark
1.2.2,
(iii);
Remark
1.5.4,
(iii);
Proposition
3.5,
(ii)
—
i.e.,
where
[cf.
the
notational
conventions
of
Propositions
3.2,
(ii);
3.5,
(ii)]
one
takes
the
S
⊆
E
of
the
present
discussion
to
be
“I((−))
⊆
I
Q
((−))”,
and
we
observe
that,
in
the
context
of
upper
semi-commutativity/semi-compatibility,
the
empty
set
always
arises
as
an
intersection
between
a
nonempty
set
and
the
domain
of
definition
[cf.
the
discussion
of
Remark
1.1.1]
of
the
“set-theoretic
logarithm
map”
under
consideration.
(vi)
Let
us
return
to
the
discussion
of
(ii),
(iii),
(iv).
Let
P
∈
P.
Then
let
us
observe
that
the
abstract
set-theoretic
“
-formalism”
of
(v)
—
i.e.,
where
one
takes
“S
⊆
E”
to
be
φ(P
)
⊆
I
Q
((−))
—
yields
a
convenient
tool
for
identifying
P
with
its
various
log-volume
hull-approximants
∈
Φ(P
)
or
∈
Ξ(P
)
[all
of
which
are
nonempty
subsets
of
φ(P
)
∈
H
—
cf.
the
discussion
of
(iii)],
i.e.,
of
passing
to
a
quotient
in
which
the
indeterminacy
discussed
in
(iii)
is
eliminated.
Moreover,
one
verifies
easily
that
this
identification
is
achieved
in
such
a
way
that
images
of
distinct
H
1
,
H
2
∈
H
map
to
the
same
subset
of
I
Q
((−))
φ(P
)
if
and
only
if
H
1
,
H
2
⊆
φ(P
).
That
is
to
say,
the
equivalence
relation
on
H
induced
by
the
quotient
map
I
Q
((−))
I
Q
((−))
φ(P
)
is
the
“expected
equivalence
relation”
H
H
1
∼
H
2
∈
H
⇐⇒
H
1
,
H
2
⊆
φ(P
)
on
H.
Finally,
we
observe
that
the
discussion
of
the
present
(vi)
may
be
applied
def
not
only
to
single
elements
P
∈
P,
but
also
to
bounded
families
of
elements
P
B
=
{P
β
}
β∈B
indexed
by
some
index
set
B
[i.e.,
collections
of
elements
P
β
∈
P
such
that
∪
β∈B
P
β
⊆
I
Q
((−))
is
relatively
compact],
by
taking
“S
⊆
E”
in
the
discussion
of
(v)
to
be
def
H
⊆
I
Q
((−))
φ(P
B
)
=
HH⊇P
β
,
∀β∈B
[cf.
the
representation
of
φ(P
)
as
an
intersection
in
(ii)].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
131
(vii)
The
operation
of
forming
the
hull
will
play
a
crucial
role
in
the
context
of
Corollary
3.12
below,
for
the
following
reason:
the
output
of
“possible
images”
[cf.
the
statement
of
Corollary
3.12]
that
arises
from
applying
the
multiradial
algorithms
of
Theorem
3.11
below
cannot
be
directly
compared
[i.e.,
at
least
in
any
a
priori
sense]
to
the
objects
in
the
local
and
global
Frobenioids
that
appear
in
the
codomain
F
×μ
-prime-strip
of
the
Θ
×μ
LGP
-link
[cf.
Definition
3.8,
(i),
(ii);
[IUTchII],
Definition
4.9,
(viii)]
determined
by
the
arithmetic
line
bundle
that
gives
rise
to
the
q-pilot
object.
The
obstructions
to
performing
such
a
comparison
may
be
eliminated
in
the
follow-
ing
way
[cf.,
especially,
the
display
of
(Ob5)]:
(Ob1)
O
×
-Indeterminacies
acting
on
tensor
packets
of
log-shells:
The
various
“possible
images”
that
occur
as
the
output
of
the
multiradial
al-
gorithms
under
consideration
are
regions
—
i.e.,
in
essence,
elements
∈
P
—
contained
in
tensor
packets
of
log-shells
I
k
[where,
for
simplicity,
we
apply
the
notational
conventions
of
Remark
1.2.2,
(i),
at
nonarchimedean
valuations].
By
contrast,
the
arithmetic
line
bundle
that
gives
rise
to
the
q-pilot
object
arises,
locally,
as
an
ideal,
i.e.,
an
O
k
-submodule,
contained
in
the
O
k
-module
O
k
,
which,
to
avoid
confusion,
we
denote
by
O
k
mdl
.
Here,
we
observe
that
unlike
the
ring
O
k
,
the
O
k
-module
O
k
mdl
does
not
admit
a
canon-
ical
generator
[i.e.,
a
canonical
element
corresponding
to
the
element
1
∈
O
k
];
by
contrast,
I
k
⊆
k
can
only
be
defined
by
using
the
ring
structure
of
O
k
and
is
not,
in
general,
stabilized
by
the
natural
action
[via
multiplication]
by
O
k
×
.
That
is
to
say,
O
k
mdl
only
admits
a
“canonical
generator”
up
to
an
inde-
terminacy
given
by
multiplication
by
O
k
×
,
i.e.,
an
indeterminacy
that
does
not
stabilize
I
k
.
(Ob2)
From
arbitrary
regions
to
arithmetic
vector
bundles,
i.e.,
hulls:
Thus,
by
passing
from
an
arbitrary
given
region
∈
P
to
the
associated
hull
φ(P
)
∈
H,
we
obtain
a
region
φ(P
)
∈
H
that
is
stabilized
by
the
natural
action
of
O
k
×
[cf.
(Ob1)]
and,
moreover,
[unlike
an
arbitrary
element
∈
P!]
may
be
regarded
as
defining
the
local
portion
of
a
global
arithmetic
vector
bundle
relative
to
the
ring
structure
labeled
by
some
α
∈
A
[i.e.,
which
is
typically
taken,
when
0
∈
A
⊆
|F
l
|,
to
be
the
zero
label
0
∈
|F
l
|].
(Ob3)
From
arithmetic
vector
bundles
to
arithmetic
line
bundles
via
“det
⊗M
(−)”:
Moreover,
by
forming
the
determinant
of
the
arithmetic
vector
bundle
constituted
by
a
hull
∈
H,
one
obtains
an
arithmetic
line
bundle,
i.e.,
which
does
indeed
yield
objects
in
the
local
and
[by
allowing
v
Q
∈
V
Q
,
v
∈
V
to
vary]
global
Frobenioids
that
appear
in
the
codomain
F
×μ
-prime-strip
of
the
Θ
×μ
LGP
-link,
hence
may
be
compared,
in
a
meaningful
way,
to
the
objects
determined
by
the
arithmetic
line
bundle
that
gives
rise
to
the
q-pilot
object.
Here,
we
observe
that:
(Ob3-1)
Weighted
tensor
powers/determinant:
In
fact,
when
form-
ing
such
a
“determinant”,
it
is
necessary
to
perform
the
following
132
SHINICHI
MOCHIZUKI
operations:
(Ob3-1-1)
In
order
to
obtain
a
“determinant”
that
is
consistent
with
the
computation
of
the
log-volume
by
means
of
certain
weighted
sums
[cf.
the
discussion
of
Remark
3.1.1],
it
is
necessary
to
work
with
suitable
positive
tensor
powers
[i.e.,
corresponding
to
the
weights
—
cf.
the
various
products
of
“N
v
’s”
in
the
discussion
of
the
final
portion
of
Remark
3.1.1,
(iv)]
of
the
deter-
minant
line
bundles
corresponding
to
the
various
direct
summands
[as
in
the
second
and
third
displays
of
Propo-
sition
3.1]
of
the
tensor
packet
of
log-shells
“I
Q
((−))”.
(Ob3-1-2)
In
order
to
obtain
a
“determinant”
that
is
consistent
with
the
normalization
of
the
log-volume
given
by
“O
(−)
”
[cf.
Proposition
3.9,
(i)],
it
is
necessary
to
tensor
the
“determinant”
of
(Ob3-1-1)
with
the
inverse
of
the
“determinant”
[in
the
sense
of
(Ob3-1-1)]
of
the
struc-
ture
sheaf
[i.e.,
“O
(−)
”],
which
may
be
thought
of
as
a
sort
of
adjustment
to
take
into
account
the
ramification
that
occurs
in
the
various
local
fields
involved.
[We
leave
the
routine
details
to
the
reader.]
(Ob3-2)
Positive
tensor
powers
of
the
determinant:
In
the
con-
text
of
(Ob3-1),
we
observe
that
there
is
no
particular
reason
to
require
that
the
various
exponents
[i.e.,
which
correspond
to
weights
—
cf.
the
various
products
of
“N
v
’s”
in
the
discussion
of
the
final
portion
of
Remark
3.1.1,
(iv)]
of
these
“suitable
positive
tensor
powers”
are
necessarily
relatively
prime.
In
particular,
the
resulting
“determinant”
might
in
fact
be
more
accurately
de-
scribed
as
a
“determinant
raised
to
some
positive
tensor
power”.
In
the
following,
we
shall
denote
this
operation
of
forming
the
“determinant
raised
to
some
positive
tensor
power”
by
means
of
the
notation
“det
⊗M
(−)”
—
where
M
denotes
the
[uniquely
determined]
positive
integer
[cf.
the
positive
integer
“N
E
”
that
appears
in
the
final
portion
of
the
discussion
of
Remark
3.1.1,
(iv)]
such
that
this
operation
“det
⊗M
(−)”
maps
[the
result
of
tensoring
the
“O
(−)
”
of
Proposi-
tion
3.9,
(i),
with]
an
arithmetic
line
bundle
to
the
M
-th
tensor
power
of
the
arithmetic
line
bundle.
Thus,
for
instance,
by
tak-
ing
M
to
be
sufficiently
large
[in
the
“multiplicative
sense”,
i.e.,
“sufficiently
divisible”],
we
may,
for
the
sake
of
simplicity,
assume
[cf.
the
“stack-theoretic
twists”
at
v
∈
V
bad
,
arising
from
the
structure
of
the
stack-theoretic
quotient
discussed
in
[IUTchI],
Remark
3.1.5]
that
the
localization
at
each
v
∈
V
of
any
arith-
metic
line
bundle
that
appears
as
the
output
of
the
operation
det
⊗M
(−)
is
always
trivial.
(Ob3-3)
Determinants
and
log-volumes:
Finally,
we
observe
in
pass-
ing
that
since
[cf.
the
situation
discussed
in
Proposition
3.9,
(iii)]
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
the
arithmetic
degree
of
such
an
arithmetic
line
bundle
may
be
in-
terpreted,
by
working
with
suitable
normalization
factors,
as
the
log-volume
of
the
original
arithmetic
vector
bundle
[i.e.,
to
which
the
operation
det
⊗M
(−)
was
applied
—
cf.
(Ob3-1),
(Ob3-2);
the
discussion
of
Remark
3.1.1],
this
intermediate
step
of
apply-
ing
det
⊗M
(−)
may
be
omitted
in
discussions
in
which
one
is
only
interested
in
computing
log-volumes.
(Ob4)
Positive
tensor
powers
of
arithmetic
line
bundles:
From
the
point
of
view
of
the
original
goal
[cf.
the
discussion
at
the
beginning
of
the
present
(vii)]
of
obtaining
objects
that
may
be
compared
to
the
objects
in
the
local
and
global
Frobenioids
that
appear
in
the
codomain
F
×μ
-
prime-strip
of
the
Θ
×μ
LGP
-link
determined
by
the
arithmetic
line
bundle
that
gives
rise
to
the
q-pilot
object,
we
thus
conclude
from
(Ob3)
that
applying
the
operation
det
⊗M
(−)
yields
objects
that
may
indeed
be
compared
to
the
objects
in
the
local
and
global
Frobenioids
that
appear
in
the
codomain
F
×μ
-prime-strip
of
the
Θ
×μ
LGP
-
link
determined
by
the
arithmetic
line
bundle
that
gives
rise
to
the
M
-th
tensor
power
of
the
q-pilot
object.
Since,
however,
the
internal
structure
of
these
local
and
global
Frobenioids
[as
well
of
as
the
localization
functors
that
relate
local
to
global
Frobe-
nioids]
remains
unaffected,
the
latter
“slightly
modified
goal”
[i.e.,
of
comparison
with
M
-th
tensor
powers
of
objects
that
arise
from
the
q-pilot
object,
as
opposed
to
the
“original
goal”
of
comparison
with
objects
that
arise
from
the
original
q-pilot
object]
does
not
result
in
any
substantive
problems
such
as,
for
instance,
an
indeterminacy
arising
from
confusion
between
a
given
arithmetic
line
bundle
and
its
M
-th
tensor
power
[i.e.,
an
indeterminacy
analogous
to
the
indeterminacy
involving
“I
ord
”
discussed
in
Remark
2.3.3,
(vi)].
One
way
to
understand
this
situation
is
as
follows:
(Ob4-1)
From
non-tensor-power
to
tensor-power
Frobenioids
via
naive
Frobenius
functors:
One
may
think
of
the
local
and
global
Frobenioids
[as
well
as
of
the
localization
functors
that
relate
local
to
global
Frobenioids]
that
appear
in
the
“slightly
modified
goal”
as
“M
-th
tensor
power
versions”
of
the
local
and
global
Frobenioids
that
appear
in
the
“original
goal”.
That
is
to
say,
one
may
think
of
these
“tensor-power
Frobenioids”
as
copies
of
the
“non-tensor-power
Frobenioids”
obtained
by
ap-
plying
the
naive
Frobenius
functor
of
degree
M
of
[FrdI],
Proposition
2.1,
(i).
In
particular,
we
conclude
[i.e.,
from
[FrdI],
Proposition
2.1,
(i)]
that
the
non-tensor-power
Frobenioids
com-
pletely
determine
the
tensor-power
Frobenioids.
(Ob4-2)
From
tensor-power
to
non-tensor-power
Frobenioids
via
tensor
power
roots:
Alternatively,
one
may
think
of
the
non-
tensor-power
Frobenioids
[i.e.,
that
appear
in
the
“original
goal”]
as
being
obtained
from
the
tensor-power
Frobenioids
[i.e.,
that
appear
in
the
“slightly
modified
goal”]
by
“extracting
M
-th
power
roots”.
Since
the
rational
function
monoids
[cf.
[FrdI],
133
134
SHINICHI
MOCHIZUKI
Theorem
5.2,
(ii)]
that
give
rise
to
the
various
local
Frobenioids
under
consideration
[cf.
[IUTchII],
Definition
4.9,
(vi),
(vii),
(viii)]
are
not
divisible
at
v
∈
V
non
,
the
tensor-power
Frobenioids
only
determine
the
non-tensor-power
Frobenioids
up
to
certain
twists.
Of
course,
these
twists
may
be
eliminated
[cf.
(Ob4-1)!]
simply
by
applying
the
naive
Frobenius
functor
of
degree
M
.
(Ob4-3)
Tensor-power-twist
indeterminacies:
In
particular,
if
one
thinks
of
the
output
of
the
crucial
operation
det
⊗M
(−)
[cf.
(Ob3)]
as
lying
in
the
tensor-power
Frobenioids,
then
one
may
always
“reconstruct”
the
non-tensor-power
Frobenioids
from
the
tensor-
power
Frobenioids
simply
by
considering
new
copies
of
the
tensor-
power
Frobenioids
which
are
related
to
the
given
copies
of
tensor-
power
Frobenioids
by
applying
the
naive
Frobenius
functor
of
de-
gree
M
whose
domain
is
the
new
copies,
and
whose
codomain
is
the
given
copies.
On
the
other
hand,
these
reconstructed
non-
tensor-power
Frobenioids,
though
completely
determined
up
to
isomorphism,
are
only
related
to
one
another,
when
regarded
over
the
given
copies
of
tensor-power
Frobenioids,
up
to
certain
twists
—
i.e.,
up
to
a
“tensor-power-twist
indeterminacy”
—
as
discussed
in
(Ob4-2).
Since,
however,
we
shall
ultimately
[e.g.,
in
the
context
of
Corollary
3.12]
only
be
interested
in
es-
timates
of
log-volumes,
such
tensor-power-twist
indeterminacies
will
not
have
any
substantive
effect
on
our
computations
[i.e.,
of
log-volumes
—
cf.
the
discussion
of
(Ob3-3)].
(Ob5)
Independence
of
the
“indeterminacy
of
possibilities”:
The
issue
of
selecting
a
specific
element
in
some
collection
of
“possible
regions”
∈
P
that
appears
in
the
output
of
the
multiradial
algorithm
is
an
issue
that
is
internal
to
the
algorithm.
In
particular,
in
order
to
compare,
in
a
meaningful
way,
the
output
of
the
algorithm
to
some
object
—
i.e.,
such
as
the
arithmetic
line
bundle
that
gives
rise
to
the
q-pilot
object
—
that
is
essentially
external
to
the
algorithm,
it
is
necessary
to
work
with
objects
that
are
independent
of
the
choice
of
such
a
specific
element/possibility.
This
may
be
achieved
by
working
with
the
hull
[cf.
the
discussion
of
(Ob1),
(Ob2),
(Ob3),
(Ob4)]
φ(P
B
)
def
associated
to
the
[bounded]
collection
of
possible
regions
P
B
=
{P
β
}
β∈B
[cf.
the
discussion
in
the
final
portion
of
(vi)]
that
appears
as
the
output
of
the
multiradial
algorithms
under
consid-
eration
and
applying
the
abstract
set-theoretic
-formalism
of
(v)
[cf.
also
(vi)].
Here,
we
observe
that
this
-formalism
of
(v)
may
be
applied
not
only
to
φ(P
B
)
but
also
[cf.
the
discussion
of
(Ob3)]
to
det
⊗M
(φ(P
B
))
and
μ
log
(φ(P
B
))
[and
in
a
compatible
fashion].
(Ob6)
Hull-approximants
for
the
log-volume
of
a
given
region:
Since
one
is
ultimately
interested
in
estimating
log-volumes
[cf.
the
discussion
of
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
135
(iii),
(iv)],
it
is
tempting
to
consider
simply
replacing
a
given
region
P
∈
P
by
μ
log
(P
).
On
the
other
hand,
in
order
to
obtain
objects
comparable
with
the
q-pilot
object
[cf.
(Ob1),
(Ob2),
(Ob3),
(Ob4),
(Ob5)],
one
is
obliged
to
work
with
hulls
∈
H
[cf.
also
the
discussion
of
(Ob7)
below].
This
state
of
affairs
suggests
working
with,
for
instance,
Ξ(P
)
⊆
H,
i.e.,
with
hull-approximants
for
μ
log
(P
)
[cf.
the
discussion
of
(iii),
(vi)].
In
this
context,
it
is
useful
to
recall
[cf.
the
discussion
of
(iv)]
that,
in
general,
it
is
not
so
clear
whether
or
not
Ξ(P
)
=
∅.
This
already
makes
it
more
natural
to
consider
Φ(P
)
⊆
H
[cf.
the
discussion
of
(iii)],
i.e.,
as
opposed
to
Ξ(P
)
⊆
H.
On
the
other
hand,
the
issue
of
independence
of
the
choice
of
a
specific
possibility
internal
to
the
algorithm
under
consideration
[cf.
(Ob5)]
already
means
that
one
must
consider
μ
log
(H
Φ(P
)
)
or
μ
log
(H
Ξ(P
)
),
as
opposed
to
μ
log
(P
),
which,
in
general,
may
be
>
μ
log
(P
)
and
indeed
=
μ
log
(φ(P
))
[cf.
the
discussion
of
(iv)].
(Ob7)
Compatibility
with
log-Kummer
correspondences:
In
(Ob6),
the
discussion
of
the
issue
of
simply
replacing
a
given
region
P
∈
P
by
μ
log
(P
)
—
i.e.,
put
another
away,
of
passing
to
the
quotient
[cf.
the
discussion
of
Remark
3.9.4,
as
well
as
of
(viii)
below]
given
by
taking
the
log-volume
—
was
subject
to
the
constraint
that
one
must
construct,
i.e.,
by
working
with
hulls
[cf.
(Ob1),
(Ob2),
(Ob3),
(Ob4),
(Ob5)],
objects
that
may
be
meaningfully
compared
to
objects
in
the
local
and
global
Frobenioids
that
appear
in
the
codomain
F
×μ
-prime-strip
of
the
Θ
×μ
LGP
-link.
This
constraint
prompts
the
following
question:
Why
is
it
that
one
cannot
simply
adopt
log-volumes
as
the
“ultimate
stage
for
comparison”
—
that
is
to
say,
without
passing
through
hulls
or
objects
in
the
local
and
global
Frobe-
nioids
referred
to
above?
At
a
more
technical
level,
this
question
may
be
reformulated
as
follows:
Why
is
it
that
one
cannot
eliminate
the
F
×μ
-prime-strip
por-
tion
[cf.
[IUTchII],
Definition
4.9,
(vii)]
—
i.e.,
in
more
concrete
terms,
for,
say,
v
∈
V
non
,
the
local
Galois
groups
“G
v
”
and
units
“O
F
×μ
”
v
[cf.
the
notation
of
the
discussion
surrounding
[IUTchI],
Fig.
I1.2;
here
and
in
the
following
discussion,
we
regard
“O
F
×μ
”
as
v
being
equipped
with
the
auxiliary
structure,
i.e.,
a
collection
of
submodules
[cf.
[IUTchII],
Definition
4.9,
(i)]
or
system
of
com-
patible
surjections
[cf.
[IUTchII],
Definition
4.9,
(v)],
with
which
it
is
equipped
in
the
definition
of
an
F
×μ
-prime-strip]
—
from
the
F
×μ
-prime-strips
that
appear
in
the
Θ
×μ
LGP
-link?
[Closely
related
issues
are
discussed
in
(ix),
(x)
below.]
The
essential
reason
for
this
may
be
understood
as
follows:
(Ob7-1)
Local
Galois
groups:
The
local
Galois
groups
“G
v
”
[for,
say,
v
∈
V
non
]
satisfy
the
important
property
of
being
invariant,
up
to
isomorphism,
with
respect
to
the
transformations
constituted
136
SHINICHI
MOCHIZUKI
by
the
Θ
×μ
LGP
-
and
log-links
—
cf.
the
vertical
and
horizontal
coricity
properties
discussed
in
Theorem
1.5,
(i),
(ii),
as
well
as
the
discussion
of
[IUTchII],
Remark
3.6.2,
(ii).
These
coricity
properties
play
a
fundamental
role
in
the
theory
of
the
present
paper,
i.e.,
by
allowing
one
to
relate,
via
these
coricity
properties,
objects
on
either
side
of
the
Θ
×μ
LGP
-
and
log-links
which
do
not
satisfy
such
invariance
properties.
In
particular,
the
theory
of
the
present
series
of
papers
cannot
function
properly
if
the
local
Galois
groups
are
eliminated
from
the
F
×μ
-prime-strips
that
appear
in
the
Θ
×μ
LGP
-link.
(Ob7-2)
Units:
Thus,
it
remains
to
consider
what
happens
if
one
elim-
inates
the
[Frobenius-like!]
units
[but
not
the
local
Galois
groups
—
cf.
(Ob7-1)!]
from
the
F
×μ
-prime-strips
that
appear
in
the
×μ
Θ
×μ
-prime-strips
LGP
-link.
This
amounts
to
replacing
the
F
×μ
that
appear
in
the
Θ
LGP
-link
by
the
associated
F
-prime-strips
[cf.
Definition
2.4,
(iii)].
Of
course,
since
one
still
has
the
local
Galois
groups,
one
can
consider
the
étale-like
units
“O
×μ
(G
v
)”
[i.e.,
“O
×μ
(G)”,
in
the
case
where
one
takes
“G”
to
be
G
v
]
of
[IUTchII],
Example
1.8,
(iv).
On
the
other
hand,
these
étale-like
units
differ
fundamentally
from
their
Frobenius-like
counter-
parts
in
the
following
respect:
·
The
Frobenius-like
units
“O
F
×μ
”
in
the
F
×μ
-
v
prime-strips
that
appear
in
the
Θ
×μ
LGP
-link
are
[tautolog-
ically!]
related
only
to
the
Frobenius-like
units
at
the
same
vertical
coordinate
[i.e.,
in
a
vertical
column
of
the
log-theta-lattice]
as
the
Θ
×μ
LGP
-link
under
consid-
eration,
i.e.,
not
to
the
Frobenius-like
units
at
other
vertical
coordinates
in
this
vertical
column.
In
particu-
lar,
these
Frobenius-like
units
arise
from
the
same
un-
derlying
multiplicative
structure
[i.e.,
of
the
ring
structure
determined,
on
various
Frobenius-like
mul-
tiplicative
monoids,
by
the
Θ
±ell
NF-Hodge
theater
to
which
they
belong]
as
the
local
and
global
[Frobenius-
like!]
value
group
portion
of
the
F
×μ
-prime-strip
under
consideration.
Put
another
way,
the
splittings
of
unit
group
and
value
group
portions
that
appear
in
the
intrinsic
structure
of
the
F
×μ
-prime-strips
un-
der
consideration
[cf.
[IUTchII],
Definition
4.9,
(vi),
(viii)]
are
consistent
with
the
underlying
multiplicative
structure
of
the
ring
structure
determined
[on
various
Frobenius-like
multiplicative
monoids]
by
the
Θ
±ell
NF-
Hodge
theater
under
consideration.
·
By
contrast,
the
étale-like
counterparts
of
these
Frobenius-like
units
are
constrained
by
their
vertical
coricity
[cf.
Theorem
1.5,
(i)]
to
be
related,
via
the
rele-
vant
log-Kummer
correspondences,
simultaneously
to
the
corresponding
Frobenius-like
units
at
every
verti-
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
137
cal
coordinate
in
a
vertical
column
of
the
log-theta-
lattice
as
the
Θ
×μ
LGP
-link
under
consideration.
In
partic-
ular,
the
relationship
between
these
étale-like
units
and
the
corresponding
Frobenius-like
units
at
various
verti-
cal
coordinates
in
the
vertical
column
under
considera-
tion
is
subject
to
the
action
of
arbitrary
iterates
of
the
log-link,
hence
to
a
complicated
confusion
between
the
unit
group
and
value
group
portions
at
various
verti-
cal
coordinates
of
this
vertical
column.
This
complicated
confusion
is
inconsistent
with
the
intrinsic
structure
of
the
F
-prime-strips
under
consideration
[cf.
Def-
inition
2.4,
(iii)],
that
is
to
say,
with
treating
the
lo-
cal
and
global
value
group
portions
of
these
F
-prime-
strips
as
objects
that
are
not
subject
to
any
constraints
in
their
relationship
to
the
étale-like
units,
i.e.,
to
the
local
Galois
group
portions
of
these
F
-prime-strips.
Put
another
way,
if
one
regards
the
étale-like
units
as
the
sole
access
route,
from
the
point
of
view
of
the
Frobenius-like
units
in
the
codomain
of
the
Θ
×μ
LGP
-link
under
consideration,
to
the
Frobenius-like
units
in
the
domain
of
this
Θ
×μ
LGP
-link,
then
one
obtains
a
situation
in
which
the
data
in
the
F
-prime-strips
[i.e.,
“non-
mutually
constrained
local/global
value
group
portions
and
local
Galois
groups]
is
“over-constrained/over-
determined”.
Thus,
in
summary,
one
cannot
eliminate
the
F
×μ
-prime-strip
portion
[cf.
[IUTchII],
Definition
4.9,
(vii)]
—
i.e.,
in
more
concrete
terms,
for,
say,
v
∈
V
non
,
the
local
Galois
groups
“G
v
”
and
units
“O
F
×μ
”
—
from
v
the
F
×μ
-prime-strips
that
appear
in
the
Θ
×μ
LGP
-link.
One
important
consequence
of
the
fact
that
the
local
Galois
group
and
unit
portions
are
indeed
included
in
the
F
×μ
-prime-strips
that
appear
in
the
Θ
×μ
LGP
-link
is
the
[“proper
functioning”,
as
described
in
the
present
paper,
of
the]
theory
of
log-Kummer
correspondences
and
log-shells
—
which
serve
as
“multiradial
containers”
[cf.
Remarks
1.5.2,
2.3.3,
2.3.4,
3.8.3]
—
both
of
which
play
a
central
role
in
the
present
paper.
(Ob8)
Vertical
shifts
in
the
output
data:
One
important
consequence
of
the
theory
of
log-Kummer
correspondences
lies
in
the
fact
that
it
allows
one
to
transport/relate
[i.e.,
by
applying
the
theory
of
log-Kummer
correspondences!]
the
output
of
the
multiradial
algorithms
under
consider-
ation
to
different
vertical
coordinates
within
a
vertical
column
of
the
log-theta-lattice.
In
fact,
this
output
—
even
if
one
works
with
hulls
[cf.
(Ob1),
(Ob2),
(Ob3),
(Ob4),
(Ob5)]!
—
yields,
a
priori,
objects
in
local
and
global
Frobenioids
that
differ,
strictly
speaking,
from
the
cor-
responding
[multiplicative!]
local
and
global
Frobenioids
that
138
SHINICHI
MOCHIZUKI
appear
in
the
input
data
of
the
algorithm
[i.e.,
the
codomain
F
×μ
-prime-strips
of
the
Θ
×μ
LGP
-link
—
cf.
Definition
3.8,
(i),
(ii)]
by
a
“vertical
shift”
in
the
log-theta-lattice,
i.e.,
more
con-
cretely,
by
an
application
of
the
log-link
[that
is
to
say,
which
produces
additive
log-shells
from
the
multiplicative
“O
×μ
’s”
in
the
input
data].
In
particular,
it
is
precisely
by
applying
the
theory
of
log-Kummer
corre-
spondences
that
we
will
ultimately
be
able
to
obtain
objects
[i.e.,
objects
in
local
and
global
Frobenioids]
arising
from
the
output
of
the
multiradial
algorithms
under
consideration
that
may
indeed
be
meaningfully
com-
pared
with
objects
in
the
local
and
global
Frobenioids
that
appear
in
the
input
data
of
the
algorithm
[cf.
Step
(xi-d)
of
the
proof
of
Corollary
3.12
below].
On
the
other
hand,
in
this
context,
it
is
important
to
note
that
since
such
comparable
objects
are
obtained
by
applying
the
log-Kummer
correspondence,
the
local
and
global
Frobenioids
to
which
these
compa-
rable
objects
belong
are
necessarily
subject
to
the
indeterminacies
of
the
relevant
log-Kummer
correspondence,
i.e.,
in
more
concrete
terms,
to
ar-
bitrary
iterates
of
the
log-link
[cf.
the
discussion
of
the
final
portion
of
Remark
3.12.2,
(v)].
(Ob9)
Hulls
in
the
context
of
the
log-link
and
log-volumes:
In
the
context
of
the
discussion
of
the
final
portion
of
(Ob8),
we
observe
that
the
operation
of
passing
to
realified
semi-simplications
[cf.
Remark
3.9.4,
(iii),
(iv),
(v),
(vi)]
in
situations
where
one
considers
the
log-link
compatibility
of
the
log-volume,
is
a
quotient
operation
on
both
the
domain
and
the
codomain
of
the
log-link
that
induces
a
natural
bijection
between
log-volumes
of
hulls
in
the
domain
and
codomain
of
the
log-
link.
That
is
to
say,
the
fact
that
this
quotient
operation
[i.e.,
of
passing
to
realified
semi-simplifications]
induces
such
a
natural
bijection
is
not
affected
—
i.e.,
unlike
the
situation
considered
in
(Ob1),
(Ob2),
(Ob3),
(Ob4),
(Ob5)!
—
by
the
fact
that
the
operation
of
passing
to
realified
semi-
simplications
[cf.
Remark
3.9.4,
(iii),
(iv),
(v),
(vi)]
involves,
at
various
intermediate
steps,
the
use
of
various
regions
which
are
not
hulls.
The
fundamental
qualitative
difference
between
the
present
situation,
on
the
one
hand,
and
the
situation
considered
in
(Ob1),
(Ob2),
(Ob3),
(Ob4),
(Ob5)
[i.e.,
which
required
the
formation
of
hulls!],
on
the
other,
may
be
understood
as
follows:
(Ob9-1)
Formal
indeterminacies
acting
on
comparable
objects:
Once
the
passage
to
comparable
objects
via
det
⊗M
(−)
of
a
suitable
hull
has
been
achieved
[cf.
the
discussion
of
(Ob5)],
the
various
formal,
or
stack-theoretic/diagram-theoretic,
indeterminacies
that
arose
from
this
passage
to
comparable
objects
—
i.e.,
·
the
tensor-power-twist
indeterminacies
of
(Ob4-3),
·
the
application
of
the
-formalism
in
(Ob5),
and
·
the
indeterminacy
with
respect
to
application
of
arbi-
trary
iterates
of
the
log-link
of
(Ob8)
—
have
no
effect
on
the
comparability
of
the
objects
obtained
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
139
in
(Ob5).
That
is
to
say,
these
indeterminacies
function
solely
as
compatibility
conditions
that
must
be
satisfied
[e.g.,
by
applying
the
theory
of
realified
semi-simplications,
as
developed
in
Remark
3.9.4,
(iii),
(iv),
(v),
(vi)]
when
passing
to
“coarse/set-theoretic
invariants”
such
as
the
log-volume.
(Ob9-2)
Non-explicit
relationships
between
comparable
and
non-
comparable
objects:
By
contrast,
the
situation
discussed
in
(Ob1),
(Ob2),
(Ob3),
(Ob4),
(Ob5)
was
one
in
which
—
until
the
“final
conclusion”
of
this
discussion
in
(Ob5)
—
compara-
ble
objects
had
not
yet
been
obtained.
Put
another
way,
prior
to
this
“final
conclusion”,
the
precise
relationship
between
the
non-comparable
objects
that
occurred
as
the
a
priori
output
of
the
multiradial
algorithms
under
consideration,
on
the
one
hand,
and
comparable
objects,
on
the
other,
had
not
yet
been
explic-
itly
computed.
Closely
related
issues
are
discussed
in
(ix)
below.
(viii)
In
the
context
of
(vi),
(vii),
it
is
of
interest
to
observe
that,
just
as
in
the
case
of
the
operations
of
(sQ1)
Kummer-detachment,
i.e.,
passing
from
Frobenius-like
[that
is
to
say,
strictly
speaking,
Frobenius-like
structures
that
contain
certain
étale-like
structures]
to
[“purely”]
étale-like
structures
[cf.
Remark
1.5.4,
(i),
as
well
as
the
vertical
arrows
of
the
commutative
diagram
of
Remark
3.10.2
below],
and
(sQ2)
Galois
evaluation
[cf.
[IUTchII],
Remark
1.12.4,
as
well
as
the
hori-
zontal
arrows
of
the
commutative
diagram
of
Remark
3.10.2
below],
the
operations
of
(sQ3)
passing
from
more
general
regions
to
positive
tensor
powers
of
de-
terminants
of
hulls
and
then
applying
the
abstract
set-theoretic
-
formalism
of
(v)
[cf.
the
discussion
of
(Ob1),
(Ob2),
(Ob3),
(Ob4),
(Ob5),
(Ob6),
(Ob7)],
(sQ4)
adjusting
the
vertical
shifts
[i.e.,
in
the
vertical
column
of
the
log-
theta-lattice
corresponding
to
the
codomain
of
the
Θ
×μ
LGP
-link
under
con-
sideration]
in
the
output
of
the
multiradial
algorithm
by
applying
the
log-Kummer
correspondence
[cf.
the
discussion
of
(Ob8)],
as
well
as
of
(sQ5)
passing
to
log-volumes
[cf.
(Ob3),
(Ob4),
(Ob6),
(Ob7),
(Ob9)],
via
the
formalism
of
realified
semi-simplifications
discussed
in
Remark
3.9.4,
may
be
regarded
as
intricate
(sub)quotient
—
or
[cf.
the
discussion
of
(ii)]
push
forward/splitting/integration
—
operations.
Indeed,
from
this
point
of
view,
the
content
of
the
entire
theory
of
the
present
series
of
papers
may
be
regarded
as
the
development
of
a
suitable
collection
of
(sub)quotient
operation
algorithms
for
con-
structing
a
140
SHINICHI
MOCHIZUKI
relatively
simple,
concrete
(sub)quotient
of
the
complicated
apparatus
constituted
by
the
full
log-theta-lattice.
The
goal
of
this
construction
of
(sub)quotient
operation
algorithms
—
i.e.,
of
the
entire
theory
of
the
present
series
of
papers
—
may
then
be
understood
as
the
computation
of
the
projection,
via
the
resulting
relatively
simple,
concrete
(sub)quotient,
of
the
“Θ-intertwining”
[i.e.,
the
structure
on
an
abstract
F
×μ
-
prime-strip
as
the
F
×μ
-prime-strip
arising
from
the
Θ-pilot
object
appearing
in
the
domain
of
the
Θ
×μ
LGP
-link
of
Definition
3.8,
(ii)]
onto
structures
arising
from
the
vertical
column
in
the
codomain
of
the
Θ
×μ
LGP
-link,
that
is
to
say,
where
the
“q-intertwining”
[i.e.,
the
structure
on
an
abstract
F
×μ
-
prime-strip
as
the
F
×μ
-prime-strip
arising
from
the
q-pilot
object
appearing
in
the
codomain
of
the
Θ
×μ
LGP
-link
of
Definition
3.8,
(ii)]
is
in
force
[cf.
the
discussion
of
Remark
3.12.2,
(ii),
below].
This
computation,
when
suitably
interpreted,
amounts,
essentially
tautologically,
to
the
inequality
of
Corollary
3.12
below.
Here,
we
observe
that
each
of
these
(sub)quotient
operations
(sQ1),
(sQ2),
(sQ3),
(sQ4),
(sQ5)
may
be
understood
as
an
operation
whose
purpose
is
to
simplify
the
quite
complicated
apparatus
con-
stituted
by
the
full
log-theta-lattice
by
allowing
the
introduction
of
various
inde-
terminacies.
Put
another
way,
the
nontriviality
of
these
various
(sub)quotient
operations
lies
in
the
very
delicate
balance
between
minimizing
the
indetermina-
cies
that
arise
from
passing
to
a
quotient,
while
at
the
same
time
ensuring
compatibility
with
the
structures
that
exist
prior
to
formation
of
the
quotient.
Indeed:
·
In
the
case
of
(sQ1),
i.e.,
the
case
of
Kummer-detachment
indetermina-
cies,
this
delicate
balance
is
discussed
in
detail
in
Remarks
1.5.4,
2.1.1,
2.2.1,
2.2.2,
2.3.3,
as
well
as
Remark
3.10.1,
(ii),
(iii),
below.
·
In
the
case
of
(sQ2),
i.e.,
the
case
of
Galois
evaluation,
the
delicate
issue
of
compatibility
with
Kummer
theory
is
discussed
in
[IUTchII],
Remark
1.12.4.
·
In
the
case
of
(sQ3),
i.e.,
the
case
of
passing
to
hulls,
various
delicate
issues
—
such
as,
for
instance,
the
delicate
issues
of
tensor-power-twist
in-
determinacies
[cf.
(Ob4-3)],
the
-formalism
[cf.
(Ob5)],
and
compatibility
with
log-Kummer
correspondences
[cf.
(Ob7)]
—
are
discussed
in
(Ob1),
(Ob2),
(Ob3),
(Ob4),
(Ob5),
(Ob6),
(Ob7)
[cf.
also
(ix),
(x)
below].
·
In
the
case
of
(sQ4),
the
adjustment
of
vertical
shifts
via
log-Kummer
correspondences
results
in
an
indeterminacy
with
respect
to
application
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
141
of
arbitrary
iterates
of
the
log-link,
i.e.,
in
the
vertical
column
of
the
log-theta-lattice
corresponding
to
the
codomain
of
the
Θ
×μ
LGP
-link
under
consideration
[cf.
(Ob8)].
·
In
the
case
of
(sQ5),
i.e.,
the
case
of
passing
to
log-volumes,
various
subtleties
surrounding
the
compatibility
of
the
log-volume
with
the
log-
link
are
discussed
in
detail
in
Remark
3.9.4,
as
well
as
in
(vii)
of
the
present
Remark
3.9.5
[cf.,
especially,
the
discussion
of
(Ob9)].
Finally,
in
this
context,
we
observe
that,
in
light
of
the
rigidity
of
étale-like
struc-
tures
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.2,
(ii)],
i.e.,
at
a
more
concrete
level,
of
objects
constructed
via
anabelian
algorithms,
the
construction
of
suit-
able
subquotients
of
the
étale-like
portion
of
the
log-theta-lattice
—
that
is
to
say,
as
in
the
case
of
(sQ2),
(sQ3),
(sQ4),
(sQ5)
—
is
a
particularly
nontrivial
issue.
(ix)
In
the
context
of
the
discussion
of
(vii)
[cf.,
especially,
(Ob7),
(Ob8),
(Ob9)],
(viii),
it
is
important
to
observe
that
there
is
a
fundamental
qualitative
difference
between
(sQ3),
(sQ4),
on
the
one
hand,
and
(sQ5),
on
the
other:
(cQ3)
Compatibility
of
(sQ3)
with
F
×μ
-prime-strip
data:
The
fact
that
[in
the
notation
of
(Ob1),
(Ob2)]
hulls
∈
H
are
stabilized
by
multiplica-
tion
by
elements
of
O
k
implies
that,
by
taking
[a
suitable
positive
tensor
power
of]
the
determinant
[cf.
(Ob3)],
they
determine
objects
[i.e.,
the
“det
⊗M
(φ(P
B
))”
of
(Ob5)]
in
the
local
and
[by
allowing
v
Q
∈
V
Q
,
v
∈
V
to
vary]
global
Frobenioids
that
appear
in
the
codomain
F
×μ
-prime-strip
of
the
Θ
×μ
LGP
-link.
In
particular,
by
considering
suitable
pull-back
mor-
phisms
in
these
local
Frobenioids
[i.e.,
which
correspond
to
base-change
morphisms
in
conventional
scheme
theory
—
cf.
[FrdI],
Definition
1.3,
(i)],
one
obtains
objects
equipped
with
natural
faithful
actions
by
the
local
Galois
groups
“G
v
”
and
units
“O
F
×μ
”
[cf.
the
notation
of
(Ob7)],
i.e.,
the
v
data
that
corresponds
to
the
F
×μ
-prime-strip
portion
of
the
F
×μ
-
prime-strips
that
appear
in
the
Θ
×μ
LGP
-link.
Moreover,
as
discussed
in
(vi),
the
quotient
induced
on
H
by
the
set-theoretic
-formalism
of
(v)
[cf.
the
display
of
(Ob5)]
may
be
understood
as
corresponding
to
the
consideration
of
the
“
-category”
consisting
of
(
lc
1
)
objects
in
the
local
Frobenioid
under
consideration
equipped
with
a
“structure
poly-morphism”
to
the
original
object
aris-
ing
from
a
hull,
i.e.,
the
“det
⊗M
(φ(P
B
))”
of
(Ob5),
given
by
the
Aut(det
⊗M
(φ(P
B
)))-orbit
of
a
linear
morphism
in
the
local
Frobenioid
[cf.
[FrdI],
Definition
1.2,
(i)]
to
det
⊗M
(φ(P
B
))
and
(
lc
2
)
morphisms
between
such
objects
that
are
compatible
with
the
structure
poly-morphism.
[Alternatively,
one
could
consider
a
slightly
modified
version
of
this
“
-
category”
by
restricting
the
objects
to
be
objects
that
arise
from
hull-
approximants
for
the
log-volume,
i.e.,
as
in
the
discussion
of
(Ob6).]
By
considering
suitable
pull-back
morphisms
in
this
-category,
we
again
ob-
tain
objects
equipped
with
mutually
compatible
[i.e.,
relative
to
varying
the
object
within
the
-category]
natural
faithful
actions
by
the
local
Galois
groups
“G
v
”
and
units
“O
F
×μ
”
[cf.
the
notation
of
(Ob7)],
i.e.,
the
data
v
142
SHINICHI
MOCHIZUKI
that
corresponds
to
the
F
×μ
-prime-strip
portion
of
the
F
×μ
-prime-
strips
that
appear
in
the
Θ
×μ
LGP
-link.
Next,
we
observe
that
one
may
consider
categories
of
“local-global
-collections
of
objects”,
i.e.,
categories
whose
objects
are
collections
consisting
of
)
a
“local”
object
in
the
-category
at
each
v
∈
V,
(
lc-gl
1
)
a
“global”
object
in
the
global
realified
Frobenioid
of
the
F
×μ
-
(
lc-gl
2
prime-strip
under
consideration,
and
)
localization
isomorphisms
between
the
image
of
the
local
object
(
lc-gl
3
at
each
v
∈
V
in
the
realification
of
the
local
Frobenioid
at
v
and
the
localization
of
the
global
object
at
the
element
∈
Prime(−)
of
the
global
realified
Frobenioid
corresponding
to
v
[and
whose
morphisms
are
compatible
collections
of
morphisms
between
the
respective
portions
of
the
data
lc-gl
and
lc-gl
]
—
cf.
the
discussion
of
1
2
the
[closely
related]
functors
in
the
final
displays
of
[FrdII],
Example
5.6,
(iii),
(iv).
In
particular,
just
as
the
tensor-power-twist
indeterminacies
of
(sQ3)
[cf.
(Ob4-3)]
and
the
indeterminacy
with
respect
to
application
of
arbitrary
iterates
of
the
log-link
of
(sQ4)
[cf.
(Ob8)]
may
be
understood
as
“formal,
or
stack-theoretic/diagram-theoretic,
quotients”
[i.e.,
as
opposed
to
“coarse/set-theoretic
quotients”
given
by
set-theoretic
in-
variants
such
as
the
log-volume
—
cf.
the
discussion
of
(Ob9-1)],
the
pair
consisting
of
(
fQ
1
)
such
a
category
of
“local-global
lc-gl
]
and
3
-collections”
[cf.
lc-gl
,
1
lc-gl
,
2
(
fQ
2
)
the
analogous
category
of
“local-global
collections”,
i.e.,
where
the
“
-category”
at
each
v
∈
V
[cf.
lc-gl
]
is
replaced
by
the
1
original
local
Frobenioid
[portion
of
the
F
×μ
-prime-strip
under
consideration]
at
each
v
∈
V,
may
also
be
regarded
as
the
“formal,
or
stack-theoretic,
quotient”
corre-
fQ
sponding
to
the
operation
of
considering
“
fQ
2
modulo
1
”.
(cQ4)
Compatibility
of
(sQ4)
with
F
×μ
-prime-strip
data:
Since
the
adjustment
of
vertical
shifts
in
(sQ4)
is
obtained
precisely
by
applying
the
log-Kummer
correspondence,
this
adjustment
operation
is
tautologically
compatible
[cf.
the
vertical
coricity
of
Theorem
1.5,
(i)]
with
suitable
iso-
morphisms
between
the
local
Galois
groups
“G
v
”
and
the
étale-like
units
“O
×μ
(G
v
)”
[cf.
the
notation
of
(Ob7-1),
(Ob7-2)]
that
appear.
Alterna-
tively,
this
adjustment
operation
is
tautologically
compatible
with
suitable
isomorphisms
between
the
local
Galois
groups
“G
v
”
and
the
Frobenius-
like
units
“O
F
×μ
”
[cf.
the
notation
of
(Ob7)],
so
long
as
one
allows
for
v
an
indeterminacy
with
respect
to
application
of
arbitrary
iterates
of
the
log-link
[cf.
the
discussion
of
(Ob8)].
(iQ5)
Incompatibility
of
(sQ5)
with
F
×μ
-prime-strip
data:
By
con-
trast,
unlike
the
situation
with
(sQ3),
(sQ4),
passing
to
log-volumes
[i.e.,
(sQ5)]
amounts
precisely
to
forgetting
the
local
Galois
groups
and
Frobenius-
like
units,
i.e.,
the
data
that
corresponds
to
the
F
×μ
-prime-strip
portion
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
143
of
the
F
×μ
-prime-strips
that
appear
in
the
Θ
×μ
LGP
-link
[cf.
the
discus-
sion
of
(Ob7)].
Here,
we
recall
that
(sQ3),
(sQ4),
(sQ5)
all
occur
within
the
vertical
column
of
the
log-theta-lattice
corresponding
to
the
codomain
of
the
Θ
×μ
LGP
-link
under
consider-
ation.
In
particular,
the
various
local
Galois
groups
“G
v
”
are
all
equipped
with
rigidifications
as
quotients
of
[isomorphs
of]
“Π
v
”
[cf.
the
notation
of
the
discussion
surrounding
[IUTchI],
Fig.
I1.2].
Put
another
way
[cf.
also
the
discussion
of
(Ob9)]:
·
One
may
think
of
the
compatibility
properties
(cQ3),
(cQ4)
as
a
sort
of
arithmetic
holomorphicity
[relative
to
the
vertical
column
under
consideration]
or,
alternatively,
as
a
sort
of
compatibility
with
the
log-
Kummer
correspondence
of
this
vertical
column.
This
point
of
view
is
reminiscent
of
the
use
of
the
descriptive
“holomorphic”
in
the
term
“holomorphic
hull”.
·
Conversely,
one
may
think
of
the
incompatibility
property
(iQ5)
as
corre-
sponding
to
the
operation
of
forgetting
this
arithmetic
holomorphic
struc-
ture
or,
alternatively,
as
a
sort
of
incompatibility
with
the
log-Kummer
correspondence
of
this
vertical
column.
From
the
point
of
view
of
the
computation
of
the
projection
of
the
Θ-intertwining
onto
the
q-intertwining
discussed
in
(viii),
this
fundamental
qualitative
difference
—
i.e.,
(cQ3),
(cQ4)
versus
(iQ5)
—
has
a
very
substantive
consequence:
It
is
precisely
by
passing
through
(sQ3),
(sQ4)
—
i.e.,
before
applying
(sQ5)!
[cf.
also
the
discussion
of
(Ob7)]
—
that
the
chain
of
poly-
isomorphisms
of
F
×μ
-prime-strips
[i.e.,
including
the
F
×μ
-prime-strip
portion
of
these
F
×μ
-prime-strips!]
that
·
begins
with
the
F
×μ
-prime-strip
arising
from
the
q-pilot
object
in
the
codomain
of
the
Θ
×μ
LGP
-link,
×μ
·
passes
through
the
Θ
×μ
LGP
-link
to
the
domain
of
the
Θ
LGP
-link,
·
passes
through
the
various
poly-isomorphisms
of
F
×μ
-prime-
strips
[cf.
the
diagram
of
Remark
3.10.2
below;
the
discussion
of
“IPL”
in
Remark
3.11.1,
(iii),
below]
induced
by
(sQ1),
(sQ2),
and
·
finally,
passes
through
(sQ3),
(sQ4),
which
satisfy
the
compat-
ibility
property
with
the
log-Kummer
correspondence
discussed
above
[i.e.,
(cQ3),
(cQ4)]
forms
a
closed
loop,
i.e.,
up
to
the
introduction
of
the
“formal
quotient
indeterminacies”
discussed
in
(cQ3),
(cQ4)
[cf.
also
the
discussion
of
(Ob9-1)].
In
this
context,
we
observe
that
a
non-closed
loop
would
yield
a
situation
from
which
no
nontrivial
conclusions
may
be
drawn,
for
essentially
the
same
reason
[that
no
nontrivial
conclusions
may
be
drawn]
as
in
the
case
of
the
“distinct
labels
approach”
of
Remark
3.11.1,
(vii),
below
[cf.
also
the
discussion
of
(Ob9-2);
Remark
3.12.2,
(ii),
(c
itw
),
(c
toy
),
below].
That
it
to
say,
it
is
only
by
constructing
such
a
closed
loop
that
one
can
complete
the
computation
of
the
projection
[that
is
to
say,
as
discussed
in
(viii)]
of
the
Θ-intertwining
onto
the
q-intertwining,
i.e.,
144
SHINICHI
MOCHIZUKI
complete
the
computation
of
the
Θ-intertwining
structure,
up
to
suitable
indeterminacies,
on
a
F
×μ
-prime-strip
that
is
constrained
to
be
subject
to
the
q-intertwining.
Here,
we
recall
from
the
discussion
of
(Ob7)
[cf.
also
(x)
below
for
a
discussion
of
a
related
topic]
that
the
construction
of
this
sort
of
mathematical
structure
—
i.e.,
a
F
×μ
-prime-strip
that
is
simultaneously
equipped
with
two
in-
tertwinings,
namely,
the
Θ-intertwining,
up
to
indeterminacies,
and
the
q-intertwining
—
cannot
be
achieved
if
one
omits
various
subportions
of
the
F
×μ
-prime-strip
por-
tion
of
the
F
×μ
-prime-strip.
It
is
this
computation/construction
that
will
allow
us,
in
Corollary
3.12
below,
to
conclude
nontrivial,
albeit
essentially
tautological,
consequences
from
the
theory
of
the
present
series
of
papers,
such
as
the
inequality
of
Corollary
3.12
[cf.
Substeps
(xi-d),
(xi-e),
(xi-f),
(xi-g)
of
the
proof
of
Corollary
3.12;
Fig.
3.8
below].
Put
another
way,
if
one
attempts
to
skip
either
(sQ3)
or
(sQ4)
and
pass
directly
from
(sQ2)
to
(sQ4)
or
(sQ5)
[or
from
(sQ3)
to
(sQ5)],
then
the
resulting
chain
of
poly-isomorphisms
of
F
×μ
-prime-strips
no
longer
forms
a
closed
loop,
and
one
can
no
longer
conclude
any
nontrivial
consequences
from
the
theory
of
the
present
series
of
papers.
(x)
In
the
context
of
the
discussion
of
(vii),
(viii),
(ix),
it
is
of
interest
to
observe
that
it
is
not
possible
[at
least
in
any
immediate
sense!]
to
work
with
regions
∈
P
that
do
not
necessarily
belong
to
H
—
and
hence
avoid
the
operation
of
passing
to
the
hull!
—
by
replacing
the
local
and
global
Frobenioids
[i.e.,
categories
of
local
and
global
arithmetic
line
bun-
dles]
that
appear
in
the
definition
of
an
F
×μ
-prime-strip
[cf.
[IUTchII],
Definition
4.9,
(viii)]
by
“more
general
categories
of
regions
∈
P”.
Indeed,
any
sort
of
category
of
regions
∈
P
necessarily
requires
consideration
of
the
multi-dimensional
underlying
space
of
I
Q
((−))
[cf.
(ii)],
i.e.,
in
essence,
an
additive
module
of
rank
>
1.
Put
another
way,
the
only
natural
way
to
relate
various
“lines”
[i.e.,
rank
1
submodules]
within
this
space
to
one
another
is
by
invoking
the
additive
structure
of
this
module.
On
the
other
hand,
since
the
Θ
×μ
LGP
-link
is
not
compatible
with
the
additive
structures
in
its
domain
and
codomain,
it
is
of
crucial
importance
that
the
categories
that
are
glued
together
via
the
Θ
×μ
LGP
-link
be
purely
multiplicative
in
nature,
i.e.,
independent,
at
least
in
an
a
priori
sense,
of
the
additive
structures
in
the
domain
and
codomain
of
the
Θ
×μ
LGP
-link.
In
particular,
one
must,
in
effect,
work
with
arithmetic
line
bundles
[which
—
unlike
arithmetic
vector
bundles
of
rank
>
1!
—
may
indeed
be
defined
in
a
way
that
only
uses
the
multiplicative
structures
of
the
rings
involved]
—
cf.
the
discussion
of
(Ob1),
(Ob2),
(Ob3),
(Ob4).
Of
course,
instead
of
working
[as
we
in
fact
do
in
the
present
series
of
papers]
with
arithmetic
line
bundles
over
F
mod
,
up
to
certain
“stack-theoretic
twists”
at
v
∈
V
bad
[cf.
[IUTchI],
Remark
3.1.5],
where
we
work
with
local
arithmetic
line
bundles
over
K
v
[which
are
necessary
in
order
to
accommodate
the
use
of
various
powers
of
q
!
—
cf.
[IUTchI],
Example
3.2,
(iv)],
v
one
could
instead
consider
working
with
arithmetic
line
bundles
over
Q.
Relative
to
the
arithmetic
line
bundles
over
F
mod
or
K
v
,
for
v
∈
V
bad
,
that
in
fact
appear
in
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
145
the
present
series
of
papers,
working
with
arithmetic
line
bundles
over
Q
amounts,
in
effect,
to
applying
some
sort
of
norm,
or
determinant
operation,
from
F
mod
down
to
Q
or,
at
v
∈
V
bad
,
from
K
v
down
to
Q
p
v
[followed
by
tensor
product
with
a
certain
fixed
arithmetic
line
bundle
on
Q
or
Q
p
v
,
in
order
to
take
into
account
the
ramification
of
F
mod
over
Q
or
K
v
over
Q
p
v
—
cf.
the
discussion
of
(Ob3-1-2)].
On
the
other
hand,
if
we
write
G
p
v
⊆
Gal(F
/Q)
for
the
unique
decomposition
group
of
p
v
that
contains
G
v
,
then
one
verifies
immediately
that
the
fact
that
G
p
v
does
not
admit
a
splitting
∼
“G
p
v
→
G
v
×
(G
p
v
/G
v
)”
implies
that
this
sort
of
norm
operation
from
K
v
down
to
Q
p
v
cannot
be
extended,
in
any
meaningful
sense,
to
any
sort
of
Galois-equivariant
[i.e.,
G
v
-equivariant]
operation
on
algebraic
closures
of
K
v
and
Q
p
v
.
Since
the
faithful
action
of
G
v
on
the
unit
portion
of
the
local
Frobenioids
in
an
F
×μ
-prime-strip
plays
a
central
role
[cf.
the
discussion
of
(Ob7)]
in
the
theory
of
log-Kummer
correspondences
and
log-shells
[which
play
a
central
role
in
the
present
paper!],
the
incompatibility
of
any
sort
of
norm
operation
with
the
local
Galois
group
G
v
makes
such
a
norm
operation
fundamentally
unsuited
for
defining
the
gluings
that
constitute
the
Θ
×μ
LGP
-
link.
Remark
3.9.6.
In
the
context
of
Proposition
3.9,
(iii),
(iv)
[cf.
also
Remark
3.9.4],
we
make
the
following
observation.
The
log-link
compatibility
of
Proposition
3.9,
(iv)
[cf.
also
Proposition
1.2,
(iii);
Proposition
1.3,
(iii);
Remark
3.9.4]
amounts
to
a
coincidence
of
log-volumes
—
not
of
arbitrary
regions
that
appear
in
the
domain
and
codomain
of
the
log-link,
but
rather
—
of
certain
types
of
“sufficiently
small”
regions
that
appear
in
the
domain
and
codomain
of
the
log-link.
On
the
other
hand,
the
“product
formula”
—
i.e.,
at
a
more
concrete
level,
the
“ratios
of
conversion”
[cf.
[IUTchI],
Remark
3.5.1,
(ii)]
between
log-volumes
at
distinct
v
∈
V
—
may
be
formulated
[without
loss
of
generality!]
in
terms
of
such
“sufficiently
small”
regions.
Thus,
in
summary,
we
conclude
that
the
log-link
compatibility
of
Proposition
3.9,
(iv),
implies
a
compati-
bility
of
“product
formulas”,
i.e.,
of
“ratios
of
conversion”
between
log-volumes
at
distinct
v
∈
V,
in
the
domain
and
codomain
of
the
log-link.
In
particular,
in
the
context
of
Proposition
3.9,
(iii),
we
conclude
that
Proposition
3.9,
(iv),
implies
a
compatibility
between
global
arithmetic
degrees
in
the
domain
and
codomain
of
the
log-link.
Remark
3.9.7.
When
computing
log-volumes
of
various
regions
of
the
sort
con-
sidered
in
Proposition
3.9,
it
is
useful
to
keep
the
following
elementary
observations
in
mind:
(i)
In
the
context
of
Proposition
3.9,
(iii),
the
defining
condition
“zero
log-
volume
for
all
but
finitely
many
v
Q
∈
V
Q
”
for
M(I
Q
(
A
F
V
Q
))
⊆
M(I
Q
(
A
F
v
Q
))
v
Q
∈V
Q
146
SHINICHI
MOCHIZUKI
that
is
imposed
on
the
various
components
indexed
by
v
Q
∈
V
Q
of
the
direct
product
of
the
above
display
may
be
satisfied
by
considering
elements
of
this
direct
product
for
which
p
w
Q
is
such
that
for
all
but
finitely
many
of
the
elements
w
Q
∈
V
non
Q
A
Q
A
unramified
in
K,
the
component
at
w
Q
is
given
by
I(
F
v
Q
)
⊆
I
(
F
v
Q
).
Indeed,
for
each
such
w
Q
∈
V
non
Q
,
the
subset
O
(−)
=
I((−))
⊆
I
Q
((−))
[cf.
the
notation
of
Proposition
3.2,
(ii);
Proposition
3.9,
(i);
the
final
sentence
of
[AbsTopIII],
Definition
5.4,
(iii)]
has
zero
log-volume.
Finally,
in
the
context
of
Proposition
3.9,
(ii),
we
observe
that,
for
each
such
w
Q
∈
V
non
Q
,
the
subset
Q
I((−))
⊆
I
((−))
is
a
mono-analytic
invariant,
which,
moreover,
[cf.
Remark
3.9.5,
(i)]
is
equal
to
its
own
holomorphic
hull.
(ii)
In
the
context
of
Proposition
3.9,
(i),
(ii),
we
observe
that
one
may
consider
the
log-volume
of
more
general,
say,
relatively
compact
subsets
E
⊆
I
Q
((−))
[cf.
the
discussion
of
Remark
3.1.1,
(iii)]
than
the
sets
which
belong
to
M(I
Q
((−))),
i.e.,
simply
by
defining
the
log-volume
of
E
to
be
the
infimum
of
the
log-
volumes
of
the
sets
E
∗
∈
M(I
Q
((−)))
such
that
E
⊆
E
∗
.
This
definition
means
that
one
must
allow
for
the
possibility
that
the
log-volume
of
E
is
−∞.
Alternatively
[and
essentially
equivalently!],
one
can
treat
such
E
by
thinking
of
such
an
E
as
corresponding
to
the
inverse
system
of
E
∗
∈
M(I
Q
((−)))
such
that
E
⊆
E
∗
.
Here,
when
E
is
a
direct
product
pre-region,
it
is
natural
to
consider
instead
the
inverse
system
of
direct
product
regions
E
∗
∈
M(I
Q
((−)))
such
that
E
⊆
E
∗
[cf.
the
discussion
of
Remark
3.1.1,
(iii)].
This
approach
via
inverse
systems
of
regions
each
of
which
has
finite
log-volume
has
the
advantage
that
it
allows
one
to
always
work
with
finite
log-volumes.
(iii)
In
a
similar
vein,
in
the
context
of
Proposition
3.9,
(iii),
we
observe
that
one
may
consider
the
log-volume
of
more
general
collections
of
relatively
compact
subsets
[cf.
the
discussion
of
Remark
3.1.1,
(iii)]
than
the
collections
of
sets
of
the
sort
considered
in
the
discussion
of
(i)
above.
Indeed,
if
{E
v
Q
⊆
I
Q
(
A
F
v
Q
)}
v
Q
∈V
Q
is
a
collection
of
subsets
such
that,
for
some
collection
of
sets
{E
v
∗
Q
}
v
Q
of
the
sort
considered
in
the
discussion
of
(i),
it
holds
that
E
v
Q
⊆
E
v
∗
Q
,
for
each
v
Q
∈
V
Q
,
then
one
may
simply
define
the
log-volume
of
{E
v
Q
}
v
Q
to
be
the
infimum
of
the
log-
volumes
of
the
collections
of
sets
{E
v
∗
Q
}
v
Q
of
the
sort
considered
in
the
discussion
of
(i)
above
such
that
E
v
Q
⊆
E
v
∗
Q
,
for
each
v
Q
∈
V
Q
[in
which
case
one
must
allow
for
the
possibility
that
the
log-volume
of
E
is
−∞];
al-
ternatively
[and
essentially
equivalently!],
one
may
think
of
such
a
collection
{E
v
Q
}
v
Q
as
corresponding
to
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
147
inverse
system
of
collections
{E
v
∗
Q
}
v
Q
of
the
sort
considered
in
the
dis-
cussion
of
(i)
above
such
that
E
v
Q
⊆
E
v
∗
Q
,
for
each
v
Q
∈
V
Q
[an
approach
that
has
the
advantage
that
it
allows
one
to
always
work
with
finite
log-volumes].
Here,
in
the
case
where
each
E
v
Q
,
for
v
Q
∈
V
Q
,
is
a
direct
product
pre-region,
it
is
natural
to
consider
instead
inverse
systems
{E
v
∗
Q
}
v
Q
as
above
such
that
each
E
v
∗
Q
,
for
v
Q
∈
V
Q
,
is
a
direct
product
region
[cf.
the
discussion
of
Remark
3.1.1,
(iii)].
Proposition
3.10.
(Global
Kummer
Theory
and
Non-interference
with
±ell
Local
Integers)
Let
{
n,m
HT
Θ
NF
}
n,m∈Z
be
a
collection
of
distinct
Θ
±ell
NF-
Hodge
theaters
[relative
to
the
given
initial
Θ-data]
—
which
we
think
of
as
aris-
ing
from
an
LGP-Gaussian
log-theta-lattice
[cf.
Definition
3.8,
(iii);
Proposi-
tion
3.5;
Remark
3.8.2].
For
each
n
∈
Z,
write
n,◦
±ell
HT
D-Θ
NF
for
the
D-Θ
±ell
NF-Hodge
theater
determined,
up
to
isomorphism,
by
the
various
±ell
n,m
HT
Θ
NF
,
where
m
∈
Z,
via
the
vertical
coricity
of
Theorem
1.5,
(i)
[cf.
Remark
3.8.2].
(i)
(Vertically
Coric
Global
LGP-,
lgp-Frobenioids
and
Associated
Kummer
Theory)
Recall
the
constructions
of
various
global
Frobenioids
in
Propo-
sition
3.7,
(i),
(ii),
(iii),
(iv),
in
the
context
of
F-prime-strip
processions.
Then
by
applying
these
constructions
to
the
F-prime-strips
“F(
n,◦
D
)
t
”
[cf.
the
notation
of
Proposition
3.5,
(i)]
and
the
various
full
log-links
associated
[cf.
the
discussion
of
Proposition
1.2,
(ix)]
to
these
F-prime-strips
—
which
we
consider
in
a
fash-
ion
compatible
with
the
F
±
l
-symmetries
involved
[cf.
Remark
1.3.2;
Propo-
sition
3.4,
(ii)]
—
we
obtain
functorial
algorithms
in
the
D-Θ
±ell
NF-Hodge
±ell
theater
n,◦
HT
D-Θ
NF
for
constructing
[number]
fields,
monoids,
and
Frobe-
nioids
equipped
with
natural
isomorphisms
±ell
M
mod
(
n,◦
HT
D-Θ
NF
±ell
n,◦
⊇
M
HT
D-Θ
mod
(
M
mod
(
n,◦
HT
D-Θ
±ell
F
mod
(
n,◦
HT
D-Θ
NF
±ell
)
α
=
M
MOD
(
n,◦
HT
D-Θ
±ell
NF
NF
NF
)
α
n,◦
)
α
=
M
HT
D-Θ
MOD
(
±ell
n,◦
)
α
⊇
M
HT
D-Θ
mod
(
∼
±ell
)
α
→
F
mod
(
n,◦
HT
D-Θ
NF
±ell
NF
NF
)
α
)
α
∼
±ell
)
α
→
F
MOD
(
n,◦
HT
D-Θ
NF
)
α
†
[cf.
the
number
fields,
monoids,
and
Frobenioids
“M
mod
(
†
D
)
j
⊇
M
mod
(
D
)
j
”,
“F
mod
(
†
D
)
j
”
of
[IUTchII],
Corollary
4.7,
(ii)]
for
α
∈
A,
where
A
is
a
subset
of
J
[cf.
Proposition
3.3],
as
well
as
F
-prime-strips
equipped
with
natural
isomor-
phisms
±ell
F
(
n,◦
HT
D-Θ
NF
∼
±ell
)
gau
→
F
(
n,◦
HT
D-Θ
NF
∼
±ell
)
LGP
→
F
(
n,◦
HT
D-Θ
NF
)
lgp
—
[all
of
]
which
we
shall
refer
to
as
being
“vertically
coric”.
For
each
n,
m
∈
Z,
these
functorial
algorithms
are
compatible
[in
the
evident
sense]
with
the
[“non-
vertically
coric”]
functorial
algorithms
of
Proposition
3.7,
(i),
(ii),
(iii),
(iv)
—
148
SHINICHI
MOCHIZUKI
i.e.,
where
[in
Proposition
3.7,
(iii),
(iv)]
we
take
“†”
to
be
“n,
m”
and
“‡”
to
be
“n,
m
−
1”
—
relative
to
the
Kummer
isomorphisms
of
labeled
data
∼
→
Ψ
cns
(
n,m
F
)
t
∼
n,◦
(
n,m
M
D
)
j
;
mod
)
j
→
M
mod
(
Ψ
cns
(
n,◦
D
)
t
∼
(
n,m
M
mod
)
j
→
M
mod
(
n,◦
D
)
j
[cf.
[IUTchII],
Corollary
4.6,
(iii);
[IUTchII],
Corollary
4.8,
(ii)]
and
the
evident
identification,
for
m
=
m,
m
−
1,
of
n,m
F
t
[i.e.,
the
F-prime-strip
that
appears
in
the
associated
Θ
±
-bridge]
with
the
F-prime-strip
associated
to
Ψ
cns
(
n,m
F
)
t
[cf.
Proposition
3.5,
(i)].
In
particular,
for
each
n,
m
∈
Z,
we
obtain
“Kummer
isomorphisms”
of
fields,
monoids,
Frobenioids,
and
F
-prime-strips
∼
(
n,m
M
mod
)
α
→
M
mod
(
n,◦
HT
D-Θ
∼
n,◦
(
n,m
M
HT
D-Θ
mod
)
α
→
M
mod
(
∼
±ell
±ell
±ell
(
n,m
F
mod
)
α
→
F
mod
(
n,◦
HT
D-Θ
∼
)
α
;
n,◦
(
n,m
M
HT
D-Θ
MOD
)
α
→
M
MOD
(
)
α
;
(
n,m
F
MOD
)
α
→
F
MOD
(
n,◦
HT
D-Θ
NF
NF
±ell
∼
±ell
NF
±ell
NF
±ell
→
F
(
n,◦
HT
D-Θ
(
n,m
M
MOD
)
α
→
M
MOD
(
n,◦
HT
D-Θ
(
n,m
F
mod
)
α
→
F
mod
(
n,◦
HT
D-Θ
n,m
F
LGP
∼
)
α
;
(
n,m
M
mod
)
α
→
M
mod
(
n,◦
HT
D-Θ
∼
NF
)
α
;
NF
)
α
;
)
LGP
;
∼
±ell
∼
±ell
∼
n,m
F
gau
n,m
F
lgp
∼
±ell
→
F
(
n,◦
HT
D-Θ
∼
±ell
→
F
(
n,◦
HT
D-Θ
NF
NF
)
α
NF
)
α
NF
±ell
n,◦
(
n,m
M
HT
D-Θ
mod
)
α
→
M
mod
(
NF
)
α
NF
)
gau
)
lgp
that
are
compatible
with
the
various
equalities,
natural
inclusions,
and
natural
isomorphisms
discussed
above.
(ii)
(Non-interference
with
Local
Integers)
In
the
notation
of
Proposi-
tions
3.2,
(ii);
3.4,
(i);
3.7,
(i),
(ii);
3.9,
(iii),
we
have
(
†
M
MOD
)
α
Ψ
log(
A,α
F
v
)
=
(
†
M
μ
MOD
)
α
v∈V
⊆
I
Q
(
A,α
F
v
)
=
I
Q
(
A
F
v
Q
)
=
I
Q
(
A
F
V
Q
)
v
Q
∈V
Q
v∈V
†
—
where
we
write
(
†
M
μ
MOD
)
α
⊆
(
M
MOD
)
α
for
the
[finite]
subgroup
of
torsion
ele-
ments,
i.e.,
roots
of
unity;
for
v
Q
∈
V
Q
,
we
identify
the
product
Vv|v
Q
I
Q
(
A,α
F
v
)
with
I
Q
(
A
F
v
Q
).
Now
let
us
think
of
the
various
groups
(
n,m
M
MOD
)
j
[of
nonzero
elements
of
a
number
field]
as
acting
on
various
portions
of
the
modules
±
I
Q
(
S
j+1
F(
n,◦
D
)
V
Q
)
[i.e.,
where
the
subscript
“V
Q
”
denote
the
direct
product
over
v
Q
∈
V
Q
—
cf.
the
notation
of
Proposition
3.5,
(ii)]
not
via
a
single
Kummer
isomorphism
as
in
(i),
but
rather
via
the
totality
of
the
various
pre-composites
of
Kummer
iso-
morphisms
with
iterates
[cf.
Remark
1.1.1]
of
the
log-links
of
the
LGP-Gaussian
)
α
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
149
log-theta-lattice
—
where
we
observe
that
these
actions
are
mutually
compatible
up
to
[harmless!]
“identity
indeterminacies”
at
an
adjacent
“m”,
precisely
as
a
consequence
of
the
equality
of
the
first
display
of
the
present
(ii)
[cf.
the
discussion
of
Remark
1.2.3,
(ii);
the
discussion
of
Definition
1.1,
(ii),
concerning
quotients
by
“Ψ
†
μ
F
N
v
”
at
v
∈
V
arc
;
the
discussion
of
Definition
1.1,
(iv),
at
v
∈
V
non
]
—
cf.
also
the
discussion
of
Remark
3.11.4
below.
Thus,
one
obtains
a
sort
of
“log-Kummer
correspondence”
between
the
totality,
as
m
ranges
over
the
elements
of
Z,
of
the
various
groups
[of
nonzero
elements
of
a
number
field]
just
discussed
[i.e.,
which
are
labeled
by
“n,
m”]
and
their
actions
[as
just
described]
on
the
“I
Q
”
labeled
by
“n,
◦”
which
is
invariant
with
respect
to
the
translation
symmetries
[cf.
Propo-
sition
1.3,
(iv)]
of
the
n-th
column
of
the
LGP-Gaussian
log-theta-lattice
[cf.
the
discussion
of
Remark
1.2.2,
(iii)].
(iii)
(Frobenioid-theoretic
log-Kummer
Correspondences)
The
relevant
Kummer
isomorphisms
of
(i)
induce,
via
the
“log-Kummer
correspondence”
of
(ii)
[cf.
also
Proposition
3.7,
(i);
Remarks
3.6.1,
3.9.2],
isomorphisms
of
Frobe-
nioids
±ell
∼
)
α
→
F
MOD
(
n,◦
HT
D-Θ
NF
)
α
(
n,m
F
MOD
∼
R
R
(
n,m
F
MOD
)
α
→
F
MOD
(
n,◦
HT
D-Θ
±ell
NF
)
α
that
are
mutually
compatible,
as
m
varies
over
the
elements
of
Z,
with
the
log-links
of
the
LGP-Gaussian
log-theta-lattice.
Moreover,
these
compatible
iso-
morphisms
of
Frobenioids,
together
with
the
relevant
Kummer
isomorphisms
of
(i),
induce,
via
the
global
“log-Kummer
correspondence”
of
(ii)
and
the
splitting
monoid
portion
of
the
“log-Kummer
correspondence”
of
Proposition
3.5,
(ii),
iso-
morphisms
of
associated
F
⊥
-prime-strips
[cf.
Definition
2.4,
(iii)]
n,m
⊥
F
LGP
∼
±ell
→
F
⊥
(
n,◦
HT
D-Θ
NF
)
LGP
that
are
mutually
compatible,
as
m
varies
over
the
elements
of
Z,
with
the
log-links
of
the
LGP-Gaussian
log-theta-lattice.
Proof.
The
various
assertions
of
Proposition
3.10
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Here,
we
observe
that
the
computation
of
the
intersection
of
the
first
display
of
(ii)
is
an
immediate
consequence
of
the
well-known
fact
that
the
set
of
nonzero
elements
of
a
number
field
that
are
integral
at
all
of
the
places
of
the
number
field
consists
of
the
set
of
roots
of
unity
contained
in
the
number
field
[cf.
the
discussion
of
Remark
1.2.3,
(ii);
[Lang],
p.
144,
the
proof
of
Theorem
5].
Remark
3.10.1.
(i)
Note
that
the
log-Kummer
correspondence
of
Proposition
3.10,
(ii),
induces
isomorphisms
of
Frobenioids
as
in
the
first
display
of
Proposition
3.10,
(iii),
precisely
)
α
”
only
involves
the
group
“(
†
M
because
the
construction
of
“(
†
F
MOD
MOD
)
α
”,
to-
gether
with
the
collection
of
subquotients
of
its
perfection
indexed
by
V
[cf.
Propo-
sition
3.7,
(i);
Remarks
3.6.1,
3.9.2].
By
contrast,
the
construction
of
“(
†
F
mod
)
α
”
also
involves
the
local
monoids
“Ψ
log(
A,α
F
v
)
⊆
log(
A,α
F
v
)”
in
an
essential
way
[cf.
150
SHINICHI
MOCHIZUKI
Proposition
3.7,
(ii)].
These
local
monoids
are
subject
to
a
somewhat
more
compli-
cated
“log-Kummer
correspondence”
[cf.
Proposition
3.5,
(ii)]
that
revolves
around
“upper
semi-compatibility”,
i.e.,
in
a
word,
one-sided
inclusions,
as
opposed
to
pre-
cise
equalities.
The
imprecise
nature
of
such
one-sided
inclusions
is
incompatible
)
α
”.
In
particular,
one
cannot
construct
log-link-
with
the
construction
of
“(
†
F
mod
compatible
isomorphisms
of
Frobenioids
for
“(
†
F
mod
)
α
”
as
in
the
first
display
of
Proposition
3.10,
(iii).
”
with
the
log-links
of
the
LGP-
(ii)
The
precise
compatibility
of
“F
MOD
Gaussian
log-theta-lattice
[cf.
the
discussion
of
(i);
the
first
“mutual
compatibil-
ity”
of
Proposition
3.10,
(iii)]
makes
it
more
suited
[i.e.,
by
comparison
to
“F
mod
”]
to
the
task
of
computing
the
Kummer-detachment
indeterminacies
[cf.
Re-
mark
1.5.4,
(i),
(iii)]
that
arise
when
one
attempts
to
pass
from
the
Frobenius-like
structures
constituted
by
the
global
portion
of
the
domain
of
the
Θ
×μ
LGP
-links
of
the
LGP-Gaussian
log-theta-lattice
to
corresponding
étale-like
structures.
That
is
to
say,
the
mutual
compatibility
of
the
isomorphisms
n,m
⊥
F
LGP
∼
±ell
→
F
⊥
(
n,◦
HT
D-Θ
NF
)
LGP
of
the
second
display
of
Proposition
3.10,
(iii),
asserts,
in
effect,
that
such
Kummer-
detachment
indeterminacies
do
not
arise.
This
is
precisely
the
reason
why
we
wish
to
work
with
the
LGP-,
as
opposed
to
the
lgp-,
Gaussian
log-theta
lattice
[cf.
Remark
3.8.1].
On
the
other
hand,
the
essentially
multiplicative
nature
of
”
[cf.
Remark
3.6.2,
(ii)]
makes
it
ill-suited
to
the
task
of
computing
the
“F
MOD
étale-transport
indeterminacies
[cf.
Remark
1.5.4,
(i),
(ii)]
that
occur
as
one
passes
between
distinct
arithmetic
holomorphic
structures
on
opposite
sides
of
a
Θ
×μ
LGP
-link.
(iii)
By
contrast,
whereas
the
additive
nature
of
the
local
modules
[i.e.,
local
”
renders
“F
mod
”
ill-suited
fractional
ideals]
that
occur
in
the
construction
of
“F
mod
to
the
computation
of
Kummer-detachment
indeterminacies
[cf.
the
discussion
of
(i),
(ii)],
the
close
relationship
[cf.
Proposition
3.9,
(i),
(ii),
(iii)]
of
these
local
mod-
ules
to
the
mono-analytic
log-shells
that
are
coric
with
respect
to
the
Θ
×μ
LGP
-link
[cf.
Theorem
1.5,
(iv);
Remark
3.8.2]
renders
“F
mod
”
well-suited
to
the
computa-
tion
of
the
étale-transport
indeterminacies
that
occur
as
one
passes
between
distinct
arithmetic
holomorphic
structures
on
opposite
sides
of
a
Θ
×μ
LGP
-link.
That
is
to
say,
although
various
distortions
of
these
local
modules
arise
as
a
result
of
both
[the
Kummer-detachment
indeterminacies
constituted
by]
the
local
“upper
semi-
compatibility”
of
Proposition
3.5,
(ii),
and
[the
étale-transport
indeterminacies
constituted
by]
the
discrepancy
between
local
holomorphic
and
mono-analytic
integral
structures
[cf.
Remark
3.9.1,
(i),
(ii)],
one
may
nevertheless
compute
—
i.e.,
if
one
takes
into
account
the
various
distortions
that
occur,
“estimate”
—
”
by
computing
log-volumes
the
global
arithmetic
degrees
of
objects
of
“F
mod
[cf.
Proposition
3.9,
(iii)],
which
are
bi-coric,
i.e.,
coric
with
respect
to
both
the
Θ
×μ
LGP
-links
[cf.
Proposition
3.9,
(ii)]
and
the
log-links
[cf.
Proposition
3.9,
(iv)]
of
the
LGP-Gaussian
log-theta-lattice.
This
computability
is
precisely
the
topic
of
Corollary
3.12
below.
On
the
other
hand,
the
issue
of
obtaining
concrete
estimates
will
be
treated
in
[IUTchIV].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
151
F
MOD
/LGP-structures
F
mod
/lgp-structures
biased
toward
multiplicative
structures
biased
toward
additive
structures
easily
related
to
value
group/non-coric
portion
“(−)
”
of
Θ
×μ
LGP
-link
easily
related
to
unit
group/coric
×μ
portion
“(−)
×μ
”
of
Θ
×μ
LGP
-/Θ
lgp
-link,
i.e.,
mono-analytic
log-shells
admits
precise
log-Kummer
correspondence
only
admits
“upper
semi-compatible”
log-Kummer
correspondence
rigid,
but
not
suited
to
explicit
computation
subject
to
substantial
distortion,
but
suited
to
explicit
estimates
Fig.
3.2:
F
MOD
/LGP-structures
versus
F
mod
/lgp-structures
(iv)
The
various
properties
of
“F
MOD
”
and
“F
mod
”
discussed
in
(i),
(ii),
(iii)
above
are
summarized
in
Fig.
3.2
above.
In
this
context,
it
is
of
interest
to
observe
that
the
natural
isomorphisms
of
Frobenioids
±ell
(
n,◦
HT
D-Θ
F
mod
NF
∼
±ell
)
α
→
F
MOD
(
n,◦
HT
D-Θ
NF
)
α
as
well
as
the
resulting
isomorphisms
of
F
-prime-strips
±ell
F
(
n,◦
HT
D-Θ
NF
∼
±ell
)
LGP
→
F
(
n,◦
HT
D-Θ
NF
)
lgp
of
Proposition
3.10,
(i),
play
the
highly
nontrivial
role
of
relating
[cf.
the
discussion
of
[IUTchII],
Remark
4.8.2,
(i)]
the
“multiplicatively
biased
F
MOD
”
to
the
“addi-
tively
biased
F
mod
”
by
means
of
the
global
ring
structure
of
the
number
field
±ell
±ell
M
mod
(
n,◦
HT
D-Θ
NF
)
α
=
M
MOD
(
n,◦
HT
D-Θ
NF
)
α
.
A
similar
statement
holds
∼
†
concerning
the
tautological
isomorphism
of
F
-prime-strips
†
F
F
lgp
of
LGP
→
Proposition
3.7,
(iv).
Remark
3.10.2.
In
the
context
of
the
various
Kummer
isomorphisms
discussed
in
the
final
display
of
Proposition
3.10,
(i),
it
is
useful
to
recall
that
the
F
×μ
-
×μ
†
×μ
prime-strips
†
F
×μ
that
appear
in
the
definition
of
the
Θ
×μ
LGP
,
F
lgp
LGP
-,
Θ
lgp
-
links
in
Definition
3.8,
(ii),
were
constructed
from
the
F
×μ
-prime-strip
†
F
×μ
env
[associated
to
the
F
-prime-strip
†
F
env
]
of
[IUTchII],
Corollary
4.10,
(ii),
in
a
152
SHINICHI
MOCHIZUKI
fashion
that
we
review
as
follows.
First,
we
remark
that,
in
the
present
discussion,
it
is
convenient
for
us
to
think
of
ourselves
as
working
with
objects
arising
from
the
LGP-Gaussian
log-theta-lattice
of
Definition
3.8,
(iii)
[so
“†”
will
be
replaced
by
“(n,
m)”
or
“(n,
◦)”].
Now
recall,
from
the
theory
developed
so
far
in
the
present
series
of
papers,
that
we
have
a
commutative
diagram
of
F
×μ
-prime-strips
∼
∼
∼
F
×μ
(
n,◦
)
env
→
F
×μ
(
n,◦
)
gau
→
F
×μ
(
n,◦
)
LGP
→
F
×μ
(
n,◦
)
lgp
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
n,m
×μ
F
env
∼
→
n,m
×μ
F
gau
∼
→
n,m
×μ
F
LGP
∼
→
n,m
×μ
F
lgp
—
where
·
for
simplicity,
we
use
the
abbreviated
version
“
n,◦
”
of
the
notation
±ell
“
n,◦
HT
D-Θ
NF
”
of
Proposition
3.10,
(i);
·
the
first
vertical
arrow
is
the
induced
F
×μ
-prime-strip
version
of
the
Kummer
isomorphism
[whose
codomain
includes
an
argument
“
†
D
>
”,
which
we
denote
here
by
“
n,◦
”]
of
the
final
display
of
Proposition
2.1,
(ii)
[cf.
also
Proposition
2.1,
(iii),
(iv),
(v)];
·
the
second,
third,
and
fourth
vertical
arrows
are
the
induced
F
×μ
-
prime-strip
versions
of
the
Kummer
isomorphisms
of
the
final
display
of
Proposition
3.10,
(i);
·
the
first
lower
horizontal
arrow
is
the
induced
F
×μ
-prime-strip
version
of
the
evaluation
isomorphism
of
the
final
display
of
[IUTchII],
Corollary
4.10,
(ii);
·
the
second
and
third
lower
horizontal
arrows
are
the
induced
F
×μ
-
prime-strip
versions
of
the
tautological
isomorphisms
of
the
final
displays
of
Proposition
3.7,
(iii),
(iv);
·
the
first
upper
horizontal
arrow
is
the
induced
F
×μ
-prime-strip
version
of
the
étale-like
evaluation
isomorphism
implicit
in
the
construction
[via
[IUTchII],
Corollary
4.6,
(iv),
(v)]
of
the
evaluation
isomorphism
of
the
final
display
of
[IUTchII],
Corollary
4.10,
(ii);
·
the
second
and
third
upper
horizontal
arrows
are
the
induced
F
×μ
-
prime-strip
versions
of
the
natural
isomorphisms
of
the
second
display
of
Proposition
3.10,
(i).
That
is
to
say,
in
summary,
n,m
×μ
the
F
×μ
-prime-strips
n,m
F
×μ
F
lgp
that
appear
in
the
Θ
×μ
LGP
,
LGP
-,
×μ
×μ
Θ
lgp
-links
of
Definition
3.8,
(iii),
were
constructed
from
the
F
-prime-
n,m
×μ
×μ
n,m
×μ
F
env
and
related
to
this
F
-prime-strip
F
env
via
the
strip
lower
horizontal
arrows
of
the
above
commutative
diagram;
moreover,
each
of
these
lower
horizontal
arrows
may
be
constructed
by
conjugating
the
corresponding
upper
horizontal
arrow
by
the
relevant
Kummer
isomor-
phisms,
i.e.,
by
the
vertical
arrows
in
the
diagram.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
153
We
are
now
ready
to
discuss
the
main
theorem
of
the
present
series
of
papers.
Theorem
3.11.
(Multiradial
Algorithms
via
LGP-Monoids/Frobenioids)
Fix
a
collection
of
initial
Θ-data
(F
/F,
X
F
,
l,
C
K
,
V,
V
bad
mod
,
)
as
in
[IUTchI],
Definition
3.1.
Let
{
n,m
HT
Θ
±ell
NF
}
n,m∈Z
be
a
collection
of
distinct
Θ
±ell
NF-Hodge
theaters
[relative
to
the
given
initial
Θ-data]
—
which
we
think
of
as
arising
from
an
LGP-Gaussian
log-theta-lattice
[cf.
Definition
3.8,
(iii)].
For
each
n
∈
Z,
write
n,◦
±ell
HT
D-Θ
NF
for
the
D-Θ
±ell
NF-Hodge
theater
determined,
up
to
isomorphism,
by
the
various
±ell
n,m
HT
Θ
NF
,
where
m
∈
Z,
via
the
vertical
coricity
of
Theorem
1.5,
(i)
[cf.
Remark
3.8.2].
(i)
(Multiradial
Representation)
Consider
the
procession
of
D
-prime-
strips
Prc(
n,◦
D
T
)
{
n,◦
D
0
}
→
{
n,◦
D
0
,
n,◦
D
1
}
→
.
.
.
→
{
n,◦
D
0
,
n,◦
D
1
,
.
.
.
,
n,◦
D
l
}
obtained
by
applying
the
natural
functor
of
[IUTchI],
Proposition
6.9,
(ii),
to
[the
±ell
D-Θ
±
-bridge
associated
to]
n,◦
HT
D-Θ
NF
.
Consider
also
the
following
data:
(a)
for
V
v
|
v
Q
∈
V
Q
,
j
∈
|F
l
|,
the
topological
modules
and
mono-
analytic
integral
structures
±
±
I(
S
j+1
;n,◦
D
v
Q
)
⊆
I
Q
(
S
j+1
;n,◦
D
v
Q
);
±
±
I(
S
j+1
,j;n,◦
D
v
)
⊆
I
Q
(
S
j+1
,j;n,◦
D
v
)
—
where
the
notation
“;
n,
◦”
denotes
the
result
of
applying
the
construc-
tion
in
question
to
the
case
of
D
-prime-strips
labeled
“n,
◦”
—
of
Proposi-
tion
3.2,
(ii)
[cf.
also
the
notational
conventions
of
Proposition
3.4,
(ii)],
which
we
regard
as
equipped
with
the
procession-normalized
mono-
analytic
log-volumes
of
Proposition
3.9,
(ii);
(b)
for
V
bad
v,
the
splitting
monoid
±ell
n,◦
HT
D-Θ
Ψ
⊥
LGP
(
NF
)
v
of
Proposition
3.5,
(ii),
(c)
[cf.
also
the
notation
of
Proposition
3.5,
(i)],
which
we
regard
—
via
the
natural
poly-isomorphisms
±
∼
±
∼
±
I
Q
(
S
j+1
,j;n,◦
D
v
)
→
I
Q
(
S
j+1
,j
F
×μ
(
n,◦
D
)
v
)
→
I
Q
(
S
j+1
,j
F(
n,◦
D
)
v
)
154
SHINICHI
MOCHIZUKI
for
j
∈
F
l
[cf.
Proposition
3.2,
(i),
(ii)]
—
as
a
subset
of
±
I
Q
(
S
j+1
,j;n,◦
D
v
)
j∈F
l
equipped
with
a(n)
[multiplicative]
action
on
±
j∈F
l
I
Q
(
S
j+1
,j;n,◦
D
v
);
(c)
for
j
∈
F
l
,
the
number
field
±ell
M
MOD
(
n,◦
HT
D-Θ
NF
±ell
)
j
=
M
mod
(
n,◦
HT
D-Θ
±
NF
⊆
I
Q
(
S
j+1
;n,◦
D
V
Q
)
=
def
)
j
±
I
Q
(
S
j+1
;n,◦
D
v
Q
)
v
Q
∈V
Q
[cf.
the
natural
poly-isomorphisms
discussed
in
(b);
Proposition
3.9,
(iii);
Proposition
3.10,
(i)],
together
with
natural
isomorphisms
between
the
associated
global
non-realified/realified
Frobenioids
±ell
F
MOD
(
n,◦
HT
D-Θ
±ell
R
F
MOD
(
n,◦
HT
D-Θ
∼
±ell
∼
±ell
NF
)
j
→
F
mod
(
n,◦
HT
D-Θ
NF
R
n,◦
)
j
→
F
mod
(
HT
D-Θ
NF
)
j
NF
)
j
[cf.
Proposition
3.10,
(i)],
whose
associated
“global
degrees”
may
be
computed
by
means
of
the
log-volumes
of
(a)
[cf.
Proposition
3.9,
(iii)].
Write
n,◦
R
LGP
for
the
collection
of
data
(a),
(b),
(c)
regarded
up
to
indeterminacies
of
the
following
two
types:
(Ind1)
the
indeterminacies
induced
by
the
automorphisms
of
the
procession
of
D
-prime-strips
Prc(
n,◦
D
T
);
(respectively,
v
Q
∈
V
arc
(Ind2)
for
each
v
Q
∈
V
non
Q
Q
),
the
indeterminacies
induced
by
the
action
of
independent
copies
of
Ism
[cf.
Proposition
1.2,
(vi)]
(respectively,
copies
of
each
of
the
automorphisms
of
order
2
whose
orbit
constitutes
the
poly-automorphism
discussed
in
Proposition
1.2,
(vii))
on
each
of
the
direct
summands
of
the
j
+1
factors
appearing
in
the
tensor
±
product
used
to
define
I
Q
(
S
j+1
;n,◦
D
v
Q
)
[cf.
(a)
above;
Proposition
3.2,
(ii)]
—
where
we
recall
that
the
cardinality
of
the
collection
of
direct
summands
is
equal
to
the
cardinality
of
the
set
of
v
∈
V
that
lie
over
v
Q
.
Then
n,◦
R
LGP
may
be
constructed
via
an
algorithm
in
the
procession
of
D
-prime-
strips
Prc(
n,◦
D
T
)
that
is
functorial
with
respect
to
isomorphisms
of
processions
of
D
-prime-strips.
For
n,
n
∈
Z,
the
permutation
symmetries
of
the
étale-
picture
discussed
in
[IUTchI],
Corollary
6.10,
(iii);
[IUTchII],
Corollary
4.11,
(ii),
(iii)
[cf.
also
Corollary
2.3,
(ii);
Remarks
2.3.2
and
3.8.2,
of
the
present
paper],
induce
compatible
poly-isomorphisms
∼
Prc(
n,◦
D
T
)
→
Prc(
n
,◦
D
T
);
n,◦
∼
R
LGP
→
n
,◦
R
LGP
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
155
which
are,
moreover,
compatible
with
the
poly-isomorphisms
n,◦
∼
D
0
→
n
,◦
D
0
induced
by
the
bi-coricity
poly-isomorphisms
of
Theorem
1.5,
(iii)
[cf.
also
[IUTchII],
Corollaries
4.10,
(iv);
4.11,
(i)].
(ii)
(log-Kummer
Correspondence)
For
n,
m
∈
Z,
the
Kummer
isomor-
phisms
of
labeled
data
∼
Ψ
cns
(
n,m
F
)
t
→
Ψ
cns
(
n,◦
D
)
t
∼
{π
1
κ-sol
(
n,m
D
)
n,m
M
}
→
{π
1
κ-sol
(
n,◦
D
)
M
(
n,◦
D
)}
j
∞
κ
j
∞
κ
∼
(
n,m
M
mod
)
j
→
M
mod
(
n,◦
D
)
j
—
where
t
∈
LabCusp
±
(
n,◦
D
)
—
of
[IUTchII],
Corollary
4.6,
(iii);
[IUTchII],
Corollary
4.8,
(i),
(ii)
[cf.
also
Propositions
3.5,
(i);
3.10,
(i),
of
the
present
paper]
induce
isomorphisms
between
the
vertically
coric
data
(a),
(b),
(c)
of
(i)
[which
we
regard,
in
the
present
(ii),
as
data
which
has
not
yet
been
subjected
to
the
indeterminacies
(Ind1),
(Ind2)
discussed
in
(i)]
and
the
corresponding
data
±ell
arising
from
each
Θ
±ell
NF-Hodge
theater
n,m
HT
Θ
NF
,
i.e.:
(a)
for
V
v
|
v
Q
,
j
∈
|F
l
|,
isomorphisms
with
local
mono-analytic
tensor
packets
and
their
Q-spans
±
∼
±
∼
±
∼
±
∼
±
∼
±
∼
±
∼
±
∼
±
)
→
I(
S
j+1
;n,◦
D
v
Q
)
I(
S
j+1
;n,m
F
v
Q
)
→
I(
S
j+1
;n,m
F
v
×μ
Q
±
)
→
I
Q
(
S
j+1
;n,◦
D
v
Q
)
I
Q
(
S
j+1
;n,m
F
v
Q
)
→
I
Q
(
S
j+1
;n,m
F
v
×μ
Q
±
I(
S
j+1
,j;n,m
F
v
)
→
I(
S
j+1
,j;n,m
F
v
×μ
)
→
I(
S
j+1
,j;n,◦
D
v
)
±
I
Q
(
S
j+1
,j;n,m
F
v
)
→
I
Q
(
S
j+1
,j;n,m
F
v
×μ
)
→
I
Q
(
S
j+1
,j;n,◦
D
v
)
[cf.
Propositions
3.2,
(i),
(ii);
3.4,
(ii);
3.5,
(i)],
all
of
which
are
com-
patible
with
the
respective
log-volumes
[cf.
Proposition
3.9,
(ii)];
(b)
for
V
bad
v,
isomorphisms
of
splitting
monoids
n,m
HT
Θ
Ψ
⊥
F
LGP
(
±ell
NF
∼
±ell
n,◦
)
v
→
Ψ
⊥
HT
D-Θ
LGP
(
NF
)
v
[cf.
Proposition
3.5,
(i);
Proposition
3.5,
(ii),
(c)];
(c)
for
j
∈
F
l
,
isomorphisms
of
number
fields
and
global
non-realified/
realified
Frobenioids
∼
±ell
(
n,m
M
MOD
)
j
→
M
MOD
(
n,◦
HT
D-Θ
∼
±ell
∼
±ell
(
n,m
F
MOD
)
j
→
F
MOD
(
n,◦
HT
D-Θ
R
R
(
n,m
F
MOD
)
j
→
F
MOD
(
n,◦
HT
D-Θ
∼
±ell
)
j
;
(
n,m
M
mod
)
j
→
M
mod
(
n,◦
HT
D-Θ
NF
)
j
;
(
n,m
F
mod
)
j
→
F
mod
(
n,◦
HT
D-Θ
NF
)
j
;
R
R
n,◦
(
n,m
F
mod
)
j
→
F
mod
(
HT
D-Θ
NF
∼
±ell
∼
±ell
NF
NF
)
j
NF
)
j
)
j
156
SHINICHI
MOCHIZUKI
which
are
compatible
with
the
respective
natural
isomorphisms
between
“MOD”-
and
“mod”-subscripted
versions
[cf.
Proposition
3.10,
(i)];
here,
the
isomorphisms
of
the
third
line
of
the
display
induce
isomorphisms
of
the
global
realified
Frobenioid
portions
n,m
C
LGP
∼
→
C
LGP
(
n,◦
HT
D-Θ
±ell
NF
);
∼
±ell
n,◦
HT
D-Θ
of
the
F
-prime-strips
n,m
F
LGP
,
F
(
±ell
F
(
n,◦
HT
D-Θ
NF
±ell
n,◦
→
C
lgp
(
HT
D-Θ
n,m
C
lgp
NF
NF
)
)
LGP
,
n,m
F
lgp
,
and
)
lgp
[cf.
Propositions
3.7,
(iii),
(iv),
(v);
3.10,
(i)].
Moreover,
as
one
varies
m
∈
Z,
the
various
isomorphisms
of
(b)
and
of
the
first
line
in
the
first
display
of
(c)
are
mutually
compatible
with
one
another,
relative
to
the
log-links
of
the
n-th
column
of
the
LGP-Gaussian
log-theta-lattice
under
consideration,
in
the
sense
that
the
only
portions
of
the
domains
of
these
isomor-
phisms
that
are
possibly
related
to
one
another
via
the
log-links
consist
of
roots
of
unity
in
the
domains
of
the
log-links
[multiplication
by
which
corresponds,
via
the
log-link,
to
an
“addition
by
zero”
indeterminacy,
i.e.,
to
no
indeterminacy!]
—
cf.
Proposition
3.5,
(ii),
(c);
Proposition
3.10,
(ii).
This
mutual
compatibility
of
the
isomorphisms
of
the
first
line
in
the
first
display
of
(c)
implies
a
corresponding
mutual
compatibility
between
the
isomorphisms
of
the
second
and
third
lines
in
the
first
display
of
(c)
that
involve
the
subscript
“MOD”
[but
not
between
the
isomorphisms
that
involve
the
subscript
“mod”!
—
cf.
Proposition
3.10,
(iii);
Re-
mark
3.10.1].
On
the
other
hand,
the
isomorphisms
of
(a)
are
subject
to
a
certain
“indeterminacy”
as
follows:
(Ind3)
as
one
varies
m
∈
Z,
the
isomorphisms
of
(a)
are
“upper
semi-
compatible”,
relative
to
the
log-links
of
the
n-th
column
of
the
LGP-
Gaussian
log-theta-lattice
under
consideration,
in
a
sense
that
involves
and
certain
natural
sur-
certain
natural
inclusions
“⊆”
at
v
Q
∈
V
non
Q
arc
jections
“”
at
v
Q
∈
V
Q
—
cf.
Proposition
3.5,
(ii),
(a),
(b),
for
more
details.
Finally,
as
one
varies
m
∈
Z,
the
isomorphisms
of
(a)
are
[precisely!]
compatible,
relative
to
the
log-links
of
the
n-th
column
of
the
LGP-Gaussian
log-theta-lattice
under
consideration,
with
the
respective
log-volumes
[cf.
Proposition
3.9,
(iv)].
(iii)
(Θ
×μ
LGP
-Link
Compatibility)
The
various
Kummer
isomorphisms
of
(ii)
satisfy
compatibility
properties
with
the
various
horizontal
arrows
—
i.e.,
Θ
×μ
LGP
-
links
—
of
the
LGP-Gaussian
log-theta-lattice
under
consideration
as
follows:
(a)
The
first
Kummer
isomorphism
of
the
first
display
of
(ii)
induces
—
by
±ell
±ell
applying
the
F
±
NF-Hodge
theater
n,m
HT
Θ
NF
l
-symmetry
of
the
Θ
∼
—
a
Kummer
isomorphism
n,m
F
×μ
→
F
×μ
(
n,◦
D
)
[cf.
The-
orem
1.5,
(iii)].
Relative
to
this
Kummer
isomorphism,
the
full
poly-
isomorphism
of
F
×μ
-prime-strips
∼
F
×μ
(
n,◦
D
)
→
F
×μ
(
n+1,◦
D
)
is
compatible
with
the
full
poly-isomorphism
of
F
×μ
-prime-strips
n,m
×μ
F
∼
→
n+1,m
×μ
F
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
157
induced
[cf.
Theorem
1.5,
(ii)]
by
the
horizontal
arrows
of
the
LGP-
Gaussian
log-theta-lattice
under
consideration
[cf.
Theorem
1.5,
(iii)].
n,◦
(b)
The
F
-prime-strips
n,m
F
D
>
)
[cf.
Proposition
2.1,
(ii)]
that
env
,
F
env
(
appear
implicitly
in
the
construction
of
the
F
-prime-strips
n,m
F
LGP
,
D-Θ
±ell
NF
D-Θ
±ell
NF
n,◦
n,m
n,◦
)
LGP
,
F
lgp
,
F
(
HT
)
lgp
[cf.
(ii),
(b),
F
(
HT
(c),
above;
Proposition
3.4,
(ii);
Proposition
3.7,
(iii),
(iv);
[IUTchII],
Corollary
4.6,
(iv),
(v);
[IUTchII],
Corollary
4.10,
(ii)]
admit
natural
∼
isomorphisms
of
associated
F
×μ
-prime-strips
n,m
F
×μ
→
n,m
F
×μ
env
,
∼
n,◦
D
>
)
[cf.
Proposition
2.1,
(vi)].
Relative
to
F
×μ
(
n,◦
D
)
→
F
×μ
env
(
these
natural
isomorphisms
and
to
the
Kummer
isomorphism
discussed
in
(a)
above,
the
full
poly-isomorphism
of
F
×μ
-prime-strips
∼
n,◦
n+1,◦
D
>
)
→
F
×μ
D
>
)
F
×μ
env
(
env
(
is
compatible
with
the
full
poly-isomorphism
of
F
×μ
-prime-strips
n,m
×μ
F
∼
→
n+1,m
×μ
F
induced
[cf.
Theorem
1.5,
(ii)]
by
the
horizontal
arrows
of
the
LGP-
Gaussian
log-theta-lattice
under
consideration
[cf.
Corollary
2.3,
(iii)].
(c)
Recall
the
data
“
n,◦
R”
[cf.
Corollary
2.3,
(ii)]
associated
to
the
D-
±ell
Θ
±ell
NF-Hodge
theater
n,◦
HT
D-Θ
NF
—
data
which
appears
implicitly
±ell
n,◦
HT
D-Θ
NF
)
LGP
,
in
the
construction
of
the
F
-prime-strips
n,m
F
LGP
,
F
(
±ell
n,m
F
lgp
,
F
(
n,◦
HT
D-Θ
NF
)
lgp
[cf.
(ii),
(b),
(c),
above;
Proposition
3.4,
(ii);
Proposition
3.7,
(iii),
(iv);
[IUTchII],
Corollary
4.6,
(iv),
(v);
±ell
[IUTchII],
Corollary
4.10,
(ii)].
This
data
that
arises
from
n,◦
HT
D-Θ
NF
is
related
to
corresponding
data
that
arises
from
the
projective
system
of
mono-theta
environments
associated
to
the
tempered
Frobenioids
of
the
±ell
Θ
±ell
NF-Hodge
theater
n,m
HT
Θ
NF
at
v
∈
V
bad
via
the
Kummer
isomorphisms
and
poly-isomorphisms
of
projective
systems
of
mono-theta
environments
discussed
in
Proposition
2.1,
(ii),
(iii)
[cf.
also
Proposition
2.1,
(vi);
the
second
display
of
Theorem
2.2,
(ii)]
and
Theorem
1.5,
(iii)
[cf.
also
(a),
(b)
above],
(v).
The
algorithmic
con-
struction
of
these
Kummer
isomorphisms
and
poly-isomorphisms
of
pro-
jective
systems
of
mono-theta
environments,
as
well
as
of
the
poly-isomor-
phism
∼
n,◦
R
→
n+1,◦
R
induced
by
any
permutation
symmetry
of
the
étale-picture
[cf.
the
fi-
±ell
nal
portion
of
(i)
above;
Corollary
2.3,
(ii);
Remark
3.8.2]
n,◦
HT
D-Θ
NF
±ell
∼
→
n+1,◦
HT
D-Θ
NF
is
compatible
with
the
horizontal
arrows
of
the
LGP-Gaussian
log-theta-lattice
under
consideration,
e.g.,
with
the
full
poly-isomorphism
of
F
×μ
-prime-strips
n,m
×μ
F
∼
→
n+1,m
×μ
F
158
SHINICHI
MOCHIZUKI
induced
[cf.
Theorem
1.5,
(ii)]
by
these
horizontal
arrows
[cf.
Corollary
2.3,
(iv)],
in
the
sense
that
these
constructions
are
stabilized/equivari-
ant/functorial
with
respect
to
arbitrary
automomorphisms
of
the
domain
and
codomain
of
these
horizontal
arrows
of
the
LGP-Gaussian
log-theta-
lattice.
Finally,
the
algorithmic
construction
of
the
poly-isomorphisms
of
the
first
display
above,
the
various
related
Kummer
isomorphisms,
and
the
various
evaluation
maps
implicit
in
the
portion
of
the
log-Kummer
correspondence
discussed
in
(ii),
(b),
are
compatible
with
the
hori-
zontal
arrows
of
the
LGP-Gaussian
log-theta-lattice
under
consideration,
i.e.,
up
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
described
in
(i),
(ii)
[cf.
also
the
discussion
of
Remark
3.11.4
below],
in
the
sense
that
these
constructions
are
stabilized/equivariant/functorial
with
respect
to
ar-
bitrary
automomorphisms
of
the
domain
and
codomain
of
these
horizontal
arrows
of
the
LGP-Gaussian
log-theta-lattice.
(d)
The
algorithmic
construction
of
the
Kummer
isomorphisms
of
the
first
display
of
(ii)
[cf.
also
(a),
(b)
above;
the
gluing
discussed
in
[IUTchII],
Corollary
4.6,
(iv);
the
Kummer
compatibilities
discussed
in
[IUTchII],
Corollary
4.8,
(iii);
the
relationship
to
the
notation
of
[IUTchI],
Definition
5.2,
(vi),
(viii),
referred
to
in
[IUTchII],
Propositions
4.2,
(i),
and
4.4,
(i)],
as
well
as
of
the
poly-isomorphisms
between
the
data
κ-sol
n,◦
(
n,◦
D
)}
j
{π
1
(
D
)
M
∞
κ
→
M
∞
κv
(
n,◦
D
v
j
)
⊆
M
∞
κ×v
(
n,◦
D
v
j
)
∼
→
v∈V
{π
1
κ-sol
(
n+1,◦
D
)
M
(
n+1,◦
D
)}
j
∞
κ
→
M
∞
κv
(
n+1,◦
D
v
j
)
⊆
M
∞
κ×v
(
n+1,◦
D
v
j
)
[i.e.,
of
the
second
line
of
the
first
display
of
[IUTchII],
Corollary
4.7,
(iii)]
induced
by
any
permutation
symmetry
of
the
étale-picture
[cf.
the
fi-
±ell
nal
portion
of
(i)
above;
Corollary
2.3,
(ii);
Remark
3.8.2]
n,◦
HT
D-Θ
NF
±ell
∼
→
n+1,◦
HT
D-Θ
NF
are
compatible
[cf.
the
discussion
of
Remark
2.3.2]
with
the
full
poly-isomorphism
of
F
×μ
-prime-strips
n,m
×μ
F
∼
→
n+1,m
×μ
F
induced
[cf.
Theorem
1.5,
(ii)]
by
the
horizontal
arrows
of
the
LGP-
Gaussian
log-theta-lattice
under
consideration,
in
the
sense
that
these
con-
structions
are
stabilized/equivariant/functorial
with
respect
to
arbi-
trary
automomorphisms
of
the
domain
and
codomain
of
these
horizon-
tal
arrows
of
the
LGP-Gaussian
log-theta-lattice.
Finally,
the
algorith-
mic
construction
of
the
poly-isomorphisms
of
the
first
display
above,
the
various
related
Kummer
isomorphisms,
and
the
various
evaluation
maps
implicit
in
the
portion
of
the
log-Kummer
correspondence
dis-
cussed
in
(ii),
(c),
are
compatible
with
the
horizontal
arrows
of
the
LGP-Gaussian
log-theta-lattice
under
consideration,
i.e.,
up
to
the
inde-
terminacies
(Ind1),
(Ind2),
(Ind3)
described
in
(i),
(ii)
[cf.
also
the
dis-
cussion
of
Remark
3.11.4
below],
in
the
sense
that
these
constructions
v∈V
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
159
are
stabilized/equivariant/functorial
with
respect
to
arbitrary
auto-
momorphisms
of
the
domain
and
codomain
of
these
horizontal
arrows
of
the
LGP-Gaussian
log-theta-lattice.
Proof.
The
various
assertions
of
Theorem
3.11
follow
immediately
from
the
defi-
nitions
and
the
references
quoted
in
the
statements
of
these
assertions
—
cf.
also
the
various
related
observations
of
Remarks
3.11.1,
3.11.2,
3.11.3,
3.11.4
below.
Remark
3.11.1.
(i)
One
way
to
summarize
the
content
of
Theorem
3.11
is
as
follows:
Theorem
3.11
gives
an
algorithm
for
describing,
up
to
certain
relatively
mild
indeterminacies,
the
LGP-monoids
[cf.
Fig.
3.1]
—
i.e.,
in
essence,
the
theta
values
2
q
j
j=1,...,l
—
which
are
constructed
relative
to
the
scheme/ring
structure,
i.e.,
“arithmetic
holomorphic
structure”,
associated
to
one
vertical
line
[i.e.,
“(n,
◦)”
for
some
fixed
n
∈
Z]
in
the
LGP-Gaussian
log-theta-lattice
under
consideration,
in
terms
of
the
a
priori
alien
arithmetic
holomorphic
structure
of
another
vertical
line
[i.e.,
“(n
+
1,
◦)”]
in
the
LGP-Gaussian
log-theta-lattice
under
consideration
[cf.,
especially,
the
final
portion
of
Theorem
3.11,
(i),
concerning
functoriality
and
compatibility
with
the
permutation
symmetries
of
the
étale-picture].
This
point
of
view
is
consistent
with
the
point
of
view
of
the
discussion
of
Remark
1.5.4;
[IUTchII],
Remark
3.8.3,
(iii).
(ii)
Although
the
various
versions
of
the
Θ-link
are
defined
[cf.
Definition
3.8,
(ii)]
as
gluings
of
the
F
×μ
-prime-strip
whose
associated
pilot
object
[cf.
[IUTchII],
Defi-
nition
4.9,
(viii)]
is
some
sort
of
Θ-pilot
object
in
the
domain
of
the
Θ-link
to
the
F
×μ
-prime-strip
whose
associated
pilot
object
is
some
sort
of
q-pilot
object
in
the
codomain
of
the
Θ-link,
in
fact
it
is
not
difficult
to
see
that
the
theory
developed
in
the
present
series
of
papers
remains
essentially
unaffected
even
if
one
replaces
this
q-pilot
F
×μ
-prime-strip
in
the
codomain
of
the
Θ-link
by
some
other
F
×μ
-prime-strip
such
as,
for
instance,
the
F
×μ
-prime-strip
whose
associated
pilot
object
is
the
q
λ
-pilot
object
[i.e.,
the
λ-th
power
of
the
q-pilot
object,
for
some
positive
integer
λ
>
1]
—
cf.
the
discussion
of
Remark
3.12.1,
(ii),
below.
One
way
to
formulate
this
observation
is
as
follows:
The
Θ-link
compatibility
described
in
Theorem
3.11,
(iii),
may
be
interpreted
as
an
assertion
to
the
effect
that
the
functorial
construction
160
SHINICHI
MOCHIZUKI
algorithm
for
the
Θ-pilot
object
up
to
certain
mild
indeterminacies
[i.e.,
(Ind1),
(Ind2),
(Ind3)]
that
is
given
in
Theorem
3.11
may
be
regarded
as
an
algorithm
whose
input
data
is
an
F
×μ
-prime-strip
[i.e.,
the
F
×μ
-prime-strip
that
appears
in
the
codomain
of
the
Θ-link],
and
whose
functoriality
is
with
respect
to
arbitrary
isomorphisms
of
the
F
×μ
-
prime-strips
that
appear
as
input
data
of
the
algorithm.
From
the
point
of
view
of
the
gluing
given
by
the
Θ-link,
this
functoriality
in
the
input
data
given
by
an
F
×μ
-prime-strip
may
be
interpreted
in
the
following
way:
this
functoriality
allows
one
to
regard
the
functorial
construction
al-
gorithm
for
the
Θ-pilot
object
up
to
certain
mild
indeterminacies
that
is
given
in
Theorem
3.11
as
an
algorithm
with
respect
to
which
the
codomain
Θ
±ell
NF-Hodge
theater
of
the
Θ-link
[together
with
the
other
Θ
±ell
NF-
Hodge
theaters
in
the
same
vertical
line
of
the
log-theta-lattice
as
this
codomain
Θ
±ell
NF-Hodge
theater]
—
i.e.,
in
effect,
the
q-pilot
F
×μ
-
prime-strip,
equipped
with
the
rigidification
determined
by
the
arith-
metic
holomorphic
structure
constituted
by
this
vertical
line
of
Θ
±ell
NF-
Hodge
theaters
—
is
“coric”,
i.e.,
“remains
invariant”/“may
be
regarded
as
being
held
fixed”
throughout
the
execution
of
the
various
operations
of
the
algorithm.
This
interpretation
will
play
a
crucial
role
in
the
application
of
Theorem
3.11
to
Corollary
3.12
below.
(iii)
On
the
other
hand,
the
étale-picture
permutation
symmetries
dis-
cussed
in
the
final
portion
of
Theorem
3.11,
(i)
[cf.
also
the
references
to
these
symmetries
in
Theorem
3.11,
(iii),
(c),
(d)],
may
be
interpreted
as
follows:
The
output
data
of
the
functorial
construction
algorithm
of
Theorem
3.11
con-
sists
of
a
representation
of
the
data
of
Theorem
3.11,
(i),
(b),
(c)
[cf.
also
Theorem
3.11,
(iii),
(c),
(d)],
up
to
certain
mild
indeterminacies
on
the
mono-analytic
étale-
like
log-shells
of
Theorem
3.11,
(i),
(a),
that
satisfies
the
following
properties:
·
(Input
prime-strip
link
(IPL))
This
output
data
is
constructed
in
such
a
way
that
it
is
linked/related,
via
full
poly-isomorphisms
of
F
×μ
-prime-strips
induced
by
operations
in
the
algorithm,
to
the
input
data
prime-strip,
i.e.,
the
“coric”/“fixed”
q-pilot
F
×μ
-
prime-strip,
equipped
with
its
rigidifying
arithmetic
holomorphic
structure
[cf.
the
discussion
of
(ii)].
In
particular,
we
note
that
each
of
these
“intermediate”
F
×μ
-prime-strips
that
appears
in
the
construc-
tion
may
itself
be
taken
to
be
both
the
input
data
of
the
functorial
algorithm
of
Theorem
3.11
[cf.
the
discussion
of
(ii)]
and
[by
applying
the
full
poly-isomorphisms
of
F
×μ
-prime-strips
that
link/relate
it
to
the
q-pilot
F
×μ
-prime-strip]
the
input
data
for
the
Kummer
theory
surrounding
the
q-pilot
object
F
×μ
-prime-strip
in
its
rigidifying
Θ
±ell
NF-Hodge
theater
[cf.
the
discussion
of
(ii)].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
161
At
a
more
explicit
level,
the
linking
isomorphisms
of
“intermediate”
F
×μ
-
prime-strips
are
given
by
composing
·
the
inverses
of
the
first
two
lower
horizontal
arrows
of
the
com-
mutative
diagram
of
Remark
3.10.2,
followed
by
·
the
first
vertical
arrow
of
this
diagram
—
corresponding
to
the
Kummer
theory
portion
of
Theorem
3.11,
(iii),
(c),
(d)
—
fol-
lowed
by
·
the
three
upper
horizontal
arrows
of
the
diagram
—
correspond-
ing
to
the
evaluation
map
portion
of
Theorem
3.11,
(iii),
(c),
(d).
Here,
we
observe
that
the
final
evaluation
map
portion
of
this
composite
involves
a
construction
of
the
Θ-pilot
object
up
to
certain
indeterminacies
[i.e.,
(Ind1),
(Ind2),
(Ind3)],
which,
by
applying
the
discussion
of
Remark
2.4.2,
(v),
(vi),
may
be
interpreted
—
provided
that
certain
sign
conditions
[cf.
the
discussion
of
Remark
2.4.2,
(iv),
(vi)]
are
satisfied,
and
one
takes
into
account
the
considerations
discussed
in
Remarks
3.9.6
[concerning
the
product
formula],
3.9.7
[concerning
inverse
systems
of
direct
product
regions]
—
as
a
construction
of
the
global
realified
Frobenioid
portion
of
an
F
×μ
-prime-strip,
together
with
various
possibilities
[corresponding
to
the
indeterminacies]
for
the
“further
rigidification”
determined
by
the
pilot
object.
·
(Simultaneous
holomorphic
expressibility
(SHE))
The
construc-
tion
of
this
output
data,
as
well
as
the
output
data
itself,
is
expressed
in
terms
that
are
simultaneously
valid/executable/well-defined
relative
to
both
the
arithmetic
holomorphic
structure
that
gives
rise
to
the
Θ-pilot
object
in
the
domain
of
the
Θ-link
—
i.e.,
in
more
technical
language,
in
terms
of/as
a
function
of
structures
in
the
Θ
±ell
NF-Hodge
theater
in
the
domain
of
the
Θ-link
—
and
the
arithmetic
holomorphic
structure
that
gives
rise
to
the
input
data
prime-strip
[i.e.,
such
as
the
q-pilot
F
×μ
-prime-
strip,
as
discussed
in
(ii)]
in
the
codomain
of
the
Θ-link
—
i.e.,
in
more
technical
language,
in
terms
of/as
a
function
of
structures
in
the
Θ
±ell
NF-Hodge
theater
in
the
codomain
of
the
Θ-link.
In
passing,
we
observe
that
this
property
“SHE”
may
be
understood,
in
a
slightly
more
concrete
way,
as
corresponding
to
the
fact
that
the
chain
of
(sub)quotients
considered
in
Remark
3.9.5,
(viii),
(ix),
forms
a
closed
loop.
These
two
fundamental
properties
of
the
output
data
of
the
algorithm
of
Theorem
3.11
will
play
a
central
role
in
the
application
of
Theorem
3.11
to
Corollary
3.12
below.
In
the
context
of
these
two
fundamental
properties,
it
is
interesting
to
ob-
serve
that,
relative
to
the
analogy
between
multiradiality
and
crystals/connections
[cf.
[IUTchII],
Remark
1.7.1;
[IUTchII],
Remark
1.9.2,
(ii),
(iii)],
162
SHINICHI
MOCHIZUKI
the
distinction
between
abstract
F
×μ
-prime-strips
and
various
specific
realizations
of
such
F
×μ
-prime-strips
[e.g.,
arising
from
the
structure
of
a
Θ
±ell
NF-Hodge
theater]
may
be
understood
as
corresponding
to
the
distinction
between
reduced
characteristic
p
schemes
[where
p
is
a
prime
number]
and
thickenings
of
such
schemes
over
Z
p
in
the
context
of
p-adic
crystals.
(iv)
The
SHE
property
discussed
in
(iii)
may
be
thought
of
as
a
sort
of
“parallel
transport”
mechanism
for
the
Θ-pilot
object
[cf.
the
analogy
between
multiradiality
and
connections,
as
discussed
in
[IUTchII],
Remark
1.7.1;
[IUTchII],
Remark
1.9.2,
(ii)],
up
to
certain
mild
indeterminacies,
from
the
[arithmetic
holomorphic
structure
represented
by
the
Θ
±ell
NF-Hodge
theater
in
the]
domain
of
the
Θ-link
to
the
[arithmetic
holomorphic
structure
represented
by
the
Θ
±ell
NF-Hodge
theater
in
the]
codomain
of
the
Θ-link.
On
the
other
hand,
in
this
context,
it
is
important
to
observe
that:
·
(Algorithmic
parallel
transport
(APT))
This
parallel
transport
mechanism
does
not
consist
of
a
simple
instance
of
transport
of
some
set-theoretic
region
[such
as
the
region
in
the
tensor
packet
of
log-shells
determined
by
the
Θ-pilot
object
in
the
domain
of
the
Θ-link]
via
some
set-theoretic
function.
Rather,
it
consists
of
a
construction
algorithm
that
is
simultaneously
valid/executable/well-defined
with
respect
to
the
arithmetic
holomorphic
structures
in
the
domain
and
codomain
of
the
Θ-link
[cf.
the
discussion
of
(iii)].
[In
this
context,
it
is
important
to
remember
that
although
this
construction
algo-
rithm
may
yield,
as
output,
various
“possible
regions”,
such
possible
regions
cannot
necessarily
be
directly
compared
with
various
structures
in
the
codomain
of
the
Θ-link.
That
is
to
say,
such
comparisons
typically
require
the
application
of
further
techniques,
as
discussed
in
Remark
3.9.5,
(vii).]
In
particular,
if
one
takes
the
point
of
view
—
as
will
be
done
in
Corollary
3.12
below!
—
that
one
is
only
interested
in
considering
the
qualitative
logical
aspects/consequences
of
the
construction
algorithm
of
Theorem
3.11,
then:
·
(Hidden
internal
structures
(HIS))
One
may
[and,
indeed,
it
is
often
useful
to]
regard
this
construction
algorithm
of
Theorem
3.11
as
a
construction
algorithm
for
producing
“some
sort
of
output
data”
satisfying
various
properties
[cf.
(iii)]
associated
to
“some
sort
of
input
data”
[cf.
(ii)]
and
forget
that
this
construction
algorithm
of
Theorem
3.11
has
anything
to
do
with
functions
[e.g.,
the
theory
of
[EtTh]]
theta
or
theta
values
[i.e.,
the
q
j
2
j=1,...,l
]!
That
is
to
say,
theta
functions/theta
values
may
be
regarded
as
HIS
of
the
con-
struction
algorithm
of
Theorem
3.11
—
somewhat
like
the
internal
structure
of
the
CPU
or
operating
system
of
a
computer!
—
i.e.,
internal
structures
whose
technical
details
are
[of
course,
of
crucial
importance
from
the
point
of
view
of
the
actual
functioning
of
the
construction
algorithm,
but
nonetheless]
irrelevant
or
uninterest-
ing
from
the
point
of
view
of
the
“end
user”,
who
is
only
interested
in
applying
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
163
construction
algorithm
to
certain
input
data
to
obtain
certain
output
data.
[Here,
we
observe
in
passing
that,
relative
to
this
analogy
with
the
internal
structure
of
the
CPU
or
operating
system
of
a
computer,
the
F
×μ
-prime-strips
that
occur
in
the
Θ-link
may
be
thought
of
as
a
sort
connecting
cable,
i.e.,
of
the
sort
that
is
used
to
link
distinct
computers
via
the
internet.
That
is
to
say,
despite
the
fact
that
such
a
connecting
cable
may
have
a
very
simple
internal
structure
by
comparison
to
the
computers
that
it
connects,
the
connection
that
it
furnishes
has
highly
nontrivial
consequences
[e.g.,
as
in
the
case
of
the
internet!]
—
cf.
the
discussion
in
(iii)
of
the
input
prime-strip
link
(IPL)
and
the
analogy
with
crystals/connections.]
On
the
other
hand,
we
observe
that,
unlike
Corollary
3.12
below,
which
only
concerns
qualitative
logical
aspects/consequences
of
the
construction
algorithm
of
Theorem
3.11,
the
explicit
computation
to
be
performed
in
[IUTchIV],
§1,
of
the
log-
volumes
that
occur
in
the
statement
of
Corollary
3.12
makes
essential
use
of
the
way
in
which
theta
values
occur
in
the
construction
algorithm
of
Theorem
3.11.
(v)
Thus,
in
summary,
the
above
discussion
yields
a
slightly
different,
and
in
some
sense
more
detailed,
way
[by
comparison
to
(i)]
to
summarize
the
content
of
the
construction
algorithm
of
Theorem
3.11
[cf.
also
the
discussion
of
Remark
3.12.2,
(ii),
below]:
The
functorial
construction
algorithm
of
Theorem
3.11
is
an
algorithm
whose
·
input
data
consists
solely
of
an
F
×μ
-prime-strip,
regarded
up
to
isomorphism
[cf.
(ii)],
and
whose
·
output
data
consists
of
certain
data
that
is
linked/related,
via
full
poly-isomorphisms
of
F
×μ
-prime-strips
induced
by
operations
in
the
algorithm,
to
the
input
data
prime-strip,
and,
moreover,
whose
con-
struction
algorithm
may
be
expressed
in
terms
that
are
simultaneously
valid/executable/well-defined
relative
to
both
the
arithmetic
holomorphic
structure
that
gives
rise
to
the
Θ-pilot
object
in
the
domain
of
the
Θ-link
and
the
arithmetic
holomorphic
structure
that
gives
rise
to
the
input
data
prime-strip
[i.e.,
such
as
the
q-pilot
F
×μ
-prime-strip].
This
construction
algorithm
of
Theorem
3.11
makes
crucial
use
of
certain
HIS
such
as
theta
functions
and
theta
values,
but
these
HIS
may
be
ignored,
if
one
is
only
interested
in
the
qualitative
logical
aspects/consequences
of
the
input
and
output
data
of
the
algorithm.
(vi)
In
the
context
of
the
input
prime-strip
link
(IPL)
and
simultaneous
holomorphic
expressibility
(SHE)
properties
discussed
in
(iii),
it
is
perhaps
of
interest
to
consider
what
happens
in
the
case
of
the
very
simple,
naive
example
discussed
in
Remark
2.2.2,
(i).
That
is
to
say,
suppose
that
one
considers
the
“naive
version”
of
the
Θ-link
given
by
a
correspondence
of
the
form
q
→
q
λ
—
where
λ
>
1
is
a
positive
integer
—
relative
to
a
single
arithmetic
holomorphic
structure,
i.e.,
in
effect,
ring
structure
“R”.
[Here,
we
remark
that,
unlike
the
situation
considered
in
the
discussion
of
(ii),
where
“q
λ
”
appears
in
the
codomain
of
some
modified
version
of
the
Θ-link,
the
“q
λ
”
in
the
present
discussion
appears
in
the
domain
of
some
modified
version
of
the
Θ-link.]
Then
the
very
definition
of
164
SHINICHI
MOCHIZUKI
this
naive
version
of
the
Θ-link
yields
an
explicit
construction
algorithm
for
“q
λ
”,
namely,
as
the
λ-th
power
of
“q”.
That
is
to
say,
this
[essentially
tautological!]
explicit
construction
algorithm
for
“q
λ
”
satisfies
the
SHE
property
considered
in
(iii)
in
the
sense
that
the
tautological
construction
algorithm
given
by
taking
“the
λ-th
power
of
q”
may
be
regarded
as
simultaneously
executable
relative
to
both
the
arithmetic
holomorphic
structure
[i.e.,
in
effect,
ring
structure]
that
gives
rise
to
“q”
and
the
arithmetic
holomorphic
structure
[i.e.,
in
effect,
ring
structure]
that
gives
rise
to
“q
λ
”.
On
the
other
hand,
we
observe
that
this
sort
of
[essentially
tautological!]
SHE
property
is
achieved
as
the
cost
of
sacrificing
the
establishment
of
the
analogue
of
the
IPL
property
of
(iii),
in
the
sense
that
if
one
restricts
oneself
to
considering
“q”
and
“q
λ
”
inside
the
fixed
con-
tainer
constituted
by
the
given
arithmetic
holomorphic
structure
[i.e.,
in
effect,
ring
structure
“R”]
that
gives
rise
to
“q”,
then
the
tauto-
logical
construction
algorithm
considered
above
does
not
induce
any
sort
of
identification
between
“q”
and
“q
λ
”.
(vii)
We
maintain
the
notation
of
(vi).
One
may
then
approach
the
issue
of
establishing
the
analogue
of
the
IPL
property
of
(iii)
by
introducing
a
formal
symbol
“∗”
[corresponding
to
the
abstract
F
×μ
-prime-strips
that
appear
in
the
Θ-link]
and
then
considering
one
of
the
following
two
approaches:
·
(Distinct
labels)
It
is
essentially
a
tautology
that
in
order
to
consider
both
of
the
assignments
∗
→
q
and
∗
→
q
λ
simultaneously
[i.e.,
in
order
to
establish
the
analogue
of
the
IPL
property
of
(iii)!],
it
is
necessary
to
introduce
distinct
labels
“†”
and
“‡”
for
the
arithmetic
holomorphic
structures
[i.e.,
in
effect,
ring
structures]
that
give
rise
to
“q”
and
“q
λ
”,
respectively.
That
is
to
say,
it
is
a
tautology
that
one
may
consider
the
assignments
∗
→
‡
q
λ
,
∗
→
†
q
simultaneously
and
without
introducing
any
inconsistencies.
On
the
other
hand,
this
approach
via
the
introduction
of
tautologically
distinct
labels
—
which
may
be
summarized
via
the
diagram
∗
→
‡
λ
q
∈
∗
‡
R
..
.
??
→
†
q
∈
..
.
†
R
IPL:
holds
SHE:
??
—
has
the
drawback
that
it
is
by
no
means
clear,
at
least
in
any
a
priori
sense,
how
to
establish
the
analogue
of
the
SHE
property
of
(iii),
since
it
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
165
is
by
no
means
clear,
at
least
in
any
a
priori
sense,
how
to
“compute”
the
relationship
between
the
“†”
and
“‡”
arithmetic
holomorphic
structures
[i.e.,
in
effect,
ring
structures].
·
(Forced
identification
of
arithmetic
holomorphic
structures)
Of
course,
one
may
then
attempt
to
remedy
the
drawback
that
appeared
in
the
distinct
labels
approach
by
simply
arbitrarily
identifying
the
“†”
and
“‡”
arithmetic
holomorphic
structures
[i.e.,
in
effect,
ring
structures],
that
is
to
say,
by
simply
deleting/forgetting
the
distinct
labels
“†”
and
“‡”.
This
approach
—
which
may
be
summarized
via
the
diagram
∗
..
.
q
λ
∈
→
??
..
.
∗
R
q
∈
→
R
IPL:
??
SHE:
holds
—
allows
one
to
apply
the
[tautological!]
construction
algorithm
discussed
in
(vi).
On
the
other
hand,
this
approach
has
the
drawback
that,
in
order
to
consider
the
assignments
∗
→
q
λ
,
∗
→
q
simultaneously
and
consistently
[i.e.,
in
order
to
establish
the
analogue
of
the
IPL
property
of
(iii)!],
one
is
led
[at
least
in
the
absence
of
more
sophisticated
machinery!]
to
regard
“q”
as
being
only
well-defined
up
n
to
possible
confusion
with
“q
λ
”,
for
some
indeterminate
n
∈
Z.
That
is
to
say,
in
summary,
this
approach
gives
rise
to
a
sort
of
“uninterest-
ing/trivial
multiradial
representation
of
“q
λ
”
via
n
“{q
λ
}
n∈Z
”
—
which
[despite
being
uninteresting/trivial!]
does
indeed
satisfy
the
formal
analogues
of
the
IPL
and
SHE
properties
of
(iii).
(viii)
We
conclude
our
discussion
of
the
simple,
naive
examples
discussed
in
(vi)
and
(vii)
by
considering
the
relationship
between
these
simple,
naive
examples
and
the
theory
of
the
present
series
of
papers.
We
begin
by
observing
that
the
n
“trivial
multiradial
representation
{q
λ
}
n∈Z
”
discussed
in
(vii)
is,
on
the
one
hand,
of
interest,
in
the
context
of
the
IPL
and
SHE
properties
of
(iii),
in
that
it
consti-
tutes
a
useful
elementary
“toy
model”
for
considering
the
qualitative
logical
aspects
of
these
fundamental
properties
satisfied
by
the
multiradial
construction
algorithm
of
Theorem
3.11.
On
the
other
hand,
this
“trivial
multiradial
represen-
tation”
is
useless
from
the
point
of
view
of
applications
such
as
the
log-volume
estimates
given
in
Corollary
3.12
below
[cf.
the
discussion
of
the
final
portion
of
(iv)]
166
SHINICHI
MOCHIZUKI
n
for
the
following
reasons:
This
“trivial
multiradial
representation
{q
λ
}
n∈Z
”
is
obtained
by
·
allowing
for
indeterminacies
in
the
value
group
portion
[i.e.,
“q
Z
”]
of
the
data
under
consideration,
·
while
the
unit
group
portion
[i.e.,
the
“O
×μ
’s”
associated
to
the
local
fields
that
appear]
of
the
data
under
consideration
is
held
rigid
[i.e.,
not
subject
to
indeterminacies];
·
only
working
with
the
multiplicative
structure
constituted
by
the
value
group
portion
of
the
rings
involved,
and
·
ignoring
issues
related
to
the
additive
structure
of
the
rings
involved,
especially,
issues
related
to
the
intertwining
between
the
additive
and
multiplicative
structures
of
these
rings
[cf.
the
discussion
of
Remark
3.12.2,
(ii),
below].
By
contrast,
the
log-volume
estimates
of
Corollary
3.12
below
rely,
in
an
essential
way,
on
the
fact
that
in
the
multiradial
construction
algorithm
of
Theorem
3.11:
·
the
value
group
portions
of
the
data
under
consideration
[i.e.,
the
F
-prime-strips
associated
to
the
F
×μ
-prime-strips
that
appear
in
the
definition
of
the
Θ-link]
are
held
rigid
[i.e.,
are
not
subject
to
inde-
terminacies],
·
while
the
unit
group
portions
of
the
data
under
consideration
[i.e.,
the
F
×μ
-prime-strips
associated
to
the
F
×μ
-prime-strips
that
appear
in
the
definition
of
the
Θ-link]
are
subject
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3);
·
the
multiradial
construction
algorithm
makes
use,
via
the
log-Kummer
correspondence,
of
the
structure
of
the
intertwining
between
the
ad-
ditive
and
multiplicative
structures
of
the
rings
involved
[cf.
the
dis-
cussion
of
Remark
3.12.2,
(ii),
(iii),
(iv),
(v),
below].
Finally,
we
observe
that
the
technique
of
assigning
distinct
labels
that
appears
in
the
distinct
labels
approach
discussed
in
(vii)
is
formalized
in
the
theory
of
the
present
series
of
papers
by
means
of
the
notion
of
Frobenius-like
structures,
i.e.,
at
a
more
concrete
level,
mathematical
objects
that,
at
least
a
priori,
only
make
sense
within
the
Θ
±ell
NF-Hodge
theater
labeled
“(n,
m)”
[where
n,
m
∈
Z]
of
the
log-theta-lattice.
The
problem
of
relating
objects
arising
from
Θ
±ell
NF-Hodge
theaters
with
distinct
labels
“(n,
m)”
is
then
resolved
in
the
present
series
of
papers
—
not
by
means
of
“forced
identification”
[i.e.,
in
the
style
of
the
discussion
of
(vii)]
of
Θ
±ell
NF-Hodge
theaters
with
distinct
labels,
but
rather
—
by
considering
the
permutation
symmetries
[i.e.,
of
the
sort
discussed
in
the
final
portion
of
Theorem
3.11,
(i)]
satisfied
by
étale-like
structures.
Here,
it
is
perhaps
useful
to
recall
that
the
fundamental
model
for
such
permutation
symmetries
is,
in
the
notation
of
[IUTchII],
Example
1.8,
(i),
Π
−→
G
←−
Π
—
where
the
arrows
“−→”
and
“←−”
denote
the
poly-morphism
given
by
compos-
∼
ing
the
natural
surjection
Π
Π/Δ
with
the
full
poly-isomorphism
Π/Δ
→
G,
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
167
and
we
observe
that
the
diagram
of
this
display
admits
a
permutation
symmetry
that
switches
these
two
arrows
“−→”
and
“←−”.
Remark
3.11.2.
(i)
In
Theorem
3.11,
(i),
we
do
not
apply
the
formalism
or
language
developed
in
[IUTchII],
§1,
for
discussing
multiradiality.
Nevertheless,
the
approach
taken
in
Theorem
3.11,
(i)
—
i.e.,
by
regarding
the
collection
of
data
(a),
(b),
(c)
up
to
the
indeterminacies
given
by
(Ind1),
(Ind2)
—
to
constructing
“multiradial
repre-
sentations”
amounts,
in
essence,
to
a
special
case
of
the
tautological
approach
to
constructing
multiradial
environments
discussed
in
[IUTchII],
Example
1.9,
(ii).
That
is
to
say,
this
tautological
approach
is
applied
to
the
vertically
coric
con-
structions
of
Proposition
3.5,
(i);
3.10,
(i),
which,
a
priori,
are
uniradial
in
the
sense
that
they
depend,
in
an
essential
way,
on
the
arithmetic
holomorphic
structure
constituted
by
a
particular
vertical
line
—
i.e.,
“(n,
◦)”
for
some
fixed
n
∈
Z
—
in
the
LGP-Gaussian
log-theta-lattice
under
consideration.
(ii)
One
important
underlying
aspect
of
the
tautological
approach
to
multira-
diality
discussed
in
(i)
is
the
treatment
of
the
various
labels
that
occur
in
the
multiplicative
and
additive
combinatorial
Teichmüller
theory
associated
to
±ell
the
D-Θ
±ell
NF-Hodge
theater
n,◦
HT
D-Θ
NF
under
consideration
[cf.
the
theory
of
[IUTchI],
§4,
§6].
The
various
transitions
between
types
of
labels
is
illustrated
in
Fig.
3.3
below.
Here,
we
recall
that:
(a)
the
passage
from
the
F
±
l
-symmetry
to
labels
∈
F
l
forms
the
content
±ell
of
the
associated
D-Θ
-Hodge
theater
[cf.
[IUTchI],
Remark
6.6.1];
(b)
the
passage
from
labels
∈
F
l
to
labels
∈
|F
l
|
forms
the
content
of
the
functorial
algorithm
of
[IUTchI],
Proposition
6.7;
(c)
the
passage
from
labels
∈
|F
l
|
to
±-processions
forms
the
content
of
[IUTchI],
Proposition
6.9,
(ii);
(d)
the
passage
from
the
F
l
-symmetry
to
labels
∈
F
l
forms
the
content
of
the
associated
D-ΘNF-Hodge
theater
[cf.
[IUTchI],
Remark
4.7.2,
(i)];
(e)
the
passage
from
labels
∈
F
l
to
-processions
forms
the
content
of
[IUTchI],
Proposition
4.11,
(ii);
(f)
the
compatibility
between
-processions
and
±-processions,
relative
to
the
natural
inclusion
of
labels
F
l
→
|F
l
|,
forms
the
content
of
[IUTchI],
Proposition
6.9,
(iii).
Here,
we
observe
in
passing
that,
in
order
to
perform
these
various
transitions,
it
is
absolutely
necessary
to
work
with
all
of
the
labels
in
F
l
or
|F
l
|,
i.e.,
one
does
not
have
the
option
of
“arbitrarily
omitting
certain
of
the
labels”
[cf.
the
discussion
of
[IUTchII],
Remark
2.6.3;
[IUTchII],
Remark
3.5.2].
Also,
in
this
context,
it
is
important
to
note
that
there
is
a
fundamental
difference
between
the
labels
∈
F
l
,
|F
l
|,
F
l
—
which
are
essentially
arithmetic
holomorphic
in
the
sense
that
they
depend,
in
an
essential
way,
on
the
various
local
and
global
arithmetic
fundamental
groups
involved
—
and
the
index
sets
of
the
mono-analytic
±-processions
168
SHINICHI
MOCHIZUKI
that
appear
in
the
multiradial
representation
of
Theorem
3.11,
(i).
Indeed,
these
index
sets
are
just
“naked
sets”
which
are
determined,
up
to
isomorphism,
by
their
cardinality.
In
particular,
the
construction
of
these
index
sets
is
independent
of
the
various
arith-
metic
holomorphic
structures
involved.
Indeed,
it
is
precisely
this
property
of
these
index
sets
that
renders
them
suitable
for
use
in
the
construction
of
the
multiradial
representations
of
Theorem
3.11,
(i).
As
discussed
in
[IUTchI],
Proposition
6.9,
(i),
for
j
∈
{0,
.
.
.
,
l
},
there
are
precisely
j
+1
possibilities
for
the
“element
labeled
j”
in
the
index
set
of
cardinality
j
+1;
this
leads
to
a
total
of
(l
+1)!
=
l
±
!
possibilities
for
the
“label
identification”
of
elements
of
index
sets
of
capsules
appearing
in
the
mono-analytic
±-processions
of
Theorem
3.11,
(i).
Finally,
in
this
context,
it
is
of
interest
to
recall
that
the
“rougher
approach
to
symmetrization”
that
arises
when
one
works
with
mono-analytic
processions
is
[“downward”]
compatible
with
the
finer
arithmetically
holomorphic
approach
to
symmetrization
that
arises
from
the
F
±
l
-symmetry
[cf.
[IUTchII],
Remark
3.5.3;
[IUTchII],
Remark
4.5.2,
(ii);
[IUTchII],
Remark
4.5.3,
(ii)].
F
±
l
-symmetry
F
l
-symmetry
⇓
⇓
labels
∈
F
l
=⇒
labels
∈
|F
l
|
⇐=
⇓
±-procession
labels
∈
F
l
⇓
⇐=
-procession
Fig.
3.3:
Transitions
from
symmetries
to
labels
to
processions
in
a
Θ
±ell
NF-Hodge
theater
(iii)
Observe
that
the
“Kummer
isomorphism
of
global
realified
Frobe-
nioids”
that
appears
in
the
theory
of
[IUTchII],
§4
—
i.e.,
more
precisely,
the
∼
various
versions
of
the
isomorphism
of
Frobenioids
“
‡
C
→
D
(
‡
D
)”
discussed
in
[IUTchII],
Corollary
4.6,
(ii),
(v)
—
is
constructed
by
considering
isomorphisms
between
local
value
groups
obtained
by
forming
the
quotient
of
the
multiplica-
tive
groups
associated
to
the
various
local
fields
that
appear
by
the
subgroups
of
local
units
[cf.
[IUTchII],
Propositions
4.2,
(ii);
4.4,
(ii)].
In
particular,
such
“Kummer
isomorphisms”
fail
to
give
rise
to
a
“log-Kummer
correspondence”,
i.e.,
they
fail
to
satisfy
mutual
compatibility
properties
of
the
sort
discussed
in
the
final
portion
of
Theorem
3.11,
(ii).
Indeed,
as
discussed
in
Remark
1.2.3,
(i)
[cf.
also
[IUTchII],
Remark
1.12.2,
(iv)],
at
v
∈
V
non
,
the
operation
of
forming
a
multi-
plicative
quotient
by
local
units
corresponds,
on
the
opposite
side
of
the
log-link,
to
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
169
forming
an
additive
quotient
by
the
submodule
obtained
as
the
p
v
-adic
logarithm
of
these
local
units.
This
is
precisely
why,
in
the
context
of
Theorem
3.11,
(ii),
we
R
wish
to
work
with
the
global
non-realified/realified
Frobenioids
“F
MOD
”,
“F
MOD
”
that
arise
from
copies
of
“F
mod
”
which
satisfy
a
“log-Kummer
correspondence”,
as
described
in
the
final
portion
of
Theorem
3.11,
(ii)
[cf.
the
discussion
of
Remark
3.10.1].
On
the
other
hand,
the
pathologies/indeterminacies
that
arise
from
work-
ing
with
global
arithmetic
line
bundles
by
means
of
various
local
data
at
v
∈
V
in
the
context
of
the
log-link
are
formalized
via
the
theory
of
the
global
Frobenioids
”,
together
with
the
“upper
semi-compatibility”
of
local
units
discussed
“F
mod
in
the
final
portion
of
Theorem
3.11,
(ii)
[cf.
also
the
discussion
of
Remark
3.10.1].
(iv)
In
the
context
of
the
discussion
of
global
realified
Frobenioids
given
in
(iii),
we
observe
that,
in
the
case
of
the
global
realified
Frobenioids
[constructed
by
means
±ell
R
n,◦
”!]
that
appear
in
the
F
-prime-strips
n,m
F
HT
D-Θ
NF
)
LGP
of
“F
MOD
LGP
,
F
(
[cf.
Theorem
3.11,
(ii),
(c)],
the
various
localization
functors
that
appear
[i.e.,
the
various
“
‡
ρ
v
”
of
[IUTchI],
Definition
5.2,
(iv);
cf.
also
the
isomorphisms
of
the
second
display
of
[IUTchII],
Corollary
4.6,
(v)]
may
be
reconstructed,
in
the
spirit
of
the
discussion
of
Remark
3.9.2,
“by
considering
the
effect
of
multiplication
by
elements
of
the
[non-realified]
global
monoids
under
consideration
on
the
log-
volumes
of
the
various
local
mono-analytic
tensor
packets
that
appear”.
[We
leave
the
routine
details
to
the
reader.]
This
reconstructibility,
together
with
the
mutual
incompatibilities
observed
in
(iii)
above
that
arise
when
one
attempts
to
work
si-
multaneously
with
log-shells
and
with
the
splitting
monoids
of
the
F
-prime-strip
n,m
F
LGP
at
v
∈
V
good
,
are
the
primary
reasons
for
our
omission
of
explicit
mention
of
the
splitting
monoids
at
v
∈
V
good
[which
in
fact
appear
as
part
of
the
data
“
n,◦
R”
considered
in
the
discussion
of
Theorem
3.11,
(iii),
(c)]
from
the
statement
of
Theorem
3.11
[cf.
Theorem
3.11,
(i),
(b);
Theorem
3.11,
(ii),
(b);
Theorem
3.11,
(iii),
(c),
in
the
case
of
v
∈
V
bad
].
Remark
3.11.3.
Before
proceeding,
we
pause
to
discuss
the
relationship
between
the
log-Kummer
correspondence
of
Theorem
3.11,
(ii),
and
the
Θ
×μ
LGP
-link
compatibility
of
Theorem
3.11,
(iii).
(i)
First,
we
recall
[cf.
Remarks
1.4.1,
(i);
3.8.2]
that
the
various
squares
that
appear
in
the
[LGP-Gaussian]
log-theta-lattice
are
far
from
being
[1-]commutative!
On
the
other
hand,
the
bi-coricity
of
F
×μ
-prime-strips
and
mono-analytic
log-
shells
discussed
in
Theorem
1.5,
(iii),
(iv),
may
be
intepreted
as
the
statement
that
the
various
squares
that
appear
in
the
[LGP-Gaussian]
log-theta-lattice
are
in
fact
[1-]commutative
with
respect
to
[the
portion
of
the
data
associated
to
each
“•”
in
the
log-theta-lattice
that
is
constituted
by]
these
bi-coric
F
×μ
-prime-strips
and
mono-analytic
log-shells.
(ii)
Next,
let
us
observe
that
in
order
to
relate
both
the
unit
and
value
group
portions
of
the
domain
and
codomain
of
the
Θ
×μ
LGP
-link
corresponding
to
adjacent
vertical
lines
—
i.e.,
(n
−
1,
∗)
and
(n,
∗)
—
of
the
[LGP-Gaussian]
log-theta-lattice
to
one
another,
it
is
necessary
to
relate
these
unit
and
value
group
portions
to
one
another
by
means
of
a
single
Θ
×μ
LGP
-link,
i.e.,
from
(n
−
1,
m)
to
(n,
m).
170
SHINICHI
MOCHIZUKI
That
is
to
say,
from
the
point
of
view
of
constructing
the
various
LGP-monoids
that
appear
in
the
multiradial
representation
of
Theorem
3.11,
(i),
one
is
tempted
to
work
with
correspondences
between
value
groups
on
adjacent
vertical
lines
that
lie
in
a
vertically
once-shifted
position
—
i.e.,
say,
at
(n
−
1,
m)
and
(n,
m)
—
relative
to
the
correspondence
between
unit
groups
on
adjacent
vertical
lines,
i.e.,
say,
at
(n
−
1,
m
−
1)
and
(n,
m
−
1).
On
the
other
hand,
such
an
approach
fails,
at
least
from
an
a
priori
point
of
view,
precisely
on
account
of
the
noncommutativity
discussed
in
(i).
Finally,
we
observe
that
in
order
to
relate
both
unit
and
value
groups
by
means
of
a
single
Θ
×μ
LGP
-link,
it
is
necessary
to
avail
oneself
of
the
Θ
×μ
LGP
-link
compatibility
properties
discussed
in
Theorem
3.11,
(iii)
—
i.e.,
of
the
theory
of
§2
and
[IUTchI],
Example
5.1,
(v);
[IUTchI],
Definition
5.2,
(vi),
(viii)
—
so
as
to
insulate
the
cyclotomes
that
appear
in
the
Kummer
theory
surrounding
the
étale
theta
function
and
κ-coric
functions
from
the
Aut
F
×μ
(−)-
indeterminacies
that
act
on
the
F
×μ
-prime-strips
involved
as
a
result
of
the
application
of
the
Θ
×μ
LGP
-link
—
cf.
the
discussion
of
Remarks
2.2.1,
2.3.2.
(iii)
As
discussed
in
(ii)
above,
a
“vertically
once-shifted”
approach
to
relating
units
on
adjacent
vertical
lines
fails
on
account
of
the
noncommutativity
discussed
in
(i).
Thus,
one
natural
approach
to
treating
the
units
in
a
“vertically
once-shifted”
fashion
—
which,
we
recall,
is
necessary
in
order
to
relate
the
LGP-monoids
on
adjacent
vertical
lines
to
one
another!
—
is
to
apply
the
bi-coricity
of
mono-
analytic
log-shells
discussed
in
(i).
On
the
other
hand,
to
take
this
approach
means
that
one
must
work
in
a
framework
that
allows
one
to
relate
[cf.
the
discussion
of
Remark
1.5.4,
(i)]
the
“Frobenius-like”
structure
constituted
by
the
Frobenioid-
theoretic
units
[i.e.,
which
occur
in
the
domain
and
codomain
of
the
Θ
×μ
LGP
-link]
to
corresponding
étale-like
structures
simultaneously
via
both
(a)
the
usual
Kummer
isomorphisms
—
i.e.,
so
as
to
be
compatible
with
the
application
of
the
compatibility
properties
of
Theorem
3.11,
(iii),
as
discussed
in
(ii)
—
and
(b)
the
composite
of
the
usual
Kummer
isomorphisms
with
[a
single
iterate
of]
the
log-link
—
i.e.,
so
as
to
be
compatible
with
the
bi-coric
treatment
of
mono-analytic
log-shells
[as
well
as
the
closely
related
construction
of
LGP-monoids]
proposed
above.
Such
a
framework
may
only
be
realized
if
one
relates
Frobenius-like
structures
to
étale-like
structures
in
a
fashion
that
is
invariant
with
respect
to
pre-composition
with
various
iterates
of
the
log-link
[cf.
the
final
portions
of
Propositions
3.5,
(ii);
3.10,
(ii)].
This
is
precisely
what
is
achieved
by
the
log-Kummer
correspondences
of
the
final
portion
of
Theorem
3.11,
(ii).
(iv)
The
discussion
of
(i),
(ii),
(iii)
above
may
be
summarized
as
follows:
The
log-Kummer
correspondences
of
the
final
portion
of
Theorem
3.11,
(ii),
allow
one
to
(a)
relate
both
the
unit
and
the
value
group
portions
of
the
domain
and
codomain
of
the
Θ
×μ
LGP
-link
corresponding
to
adjacent
vertical
lines
of
the
[LGP-Gaussian]
log-theta-lattice
to
one
another,
in
a
fashion
that
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
171
(b)
insulates
the
cyclotomes/Kummer
theory
surrounding
the
étale
theta
function
and
κ-coric
functions
involved
from
the
Aut
F
×μ
(−)-
indeterminacies
that
act
on
the
F
×μ
-prime-strips
involved
as
a
result
of
the
application
of
the
Θ
×μ
LGP
-link
[cf.
Theorem
3.11,
(iii)],
and,
moreover,
(c)
is
compatible
with
the
bi-coricity
of
the
mono-analytic
log-shells
[cf.
Theorem
1.5,
(iv)],
hence
also
with
the
operation
of
relating
the
LGP-monoids
that
appear
in
the
multiradial
representation
of
Theorem
3.11,
(i),
corresponding
to
adjacent
vertical
lines
of
the
[LGP-Gaussian]
log-theta-lattice
to
one
another.
These
observations
will
play
a
key
role
in
the
proof
of
Corollary
3.12
below.
Remark
3.11.4.
In
the
context
of
the
compatibility
discussed
in
the
final
portion
of
Theorem
3.11,
(iii),
(c),
(d),
we
make
the
following
observations.
(i)
First
of
all,
we
observe
that
consideration
of
the
log-Kummer
corre-
spondence
in
the
context
of
the
compatibility
discussed
in
the
final
portion
of
Theorem
3.11,
(iii),
(c),
(d),
amounts
precisely
to
forgetting
the
labels
of
the
various
Frobenius-like
“•’s”
[cf.
the
notation
of
the
final
display
of
Proposition
1.3,
(iv)],
i.e.,
to
identifying
data
associated
to
these
Frobenius-like
“•’s”
with
the
corresponding
data
associated
to
the
étale-like
“◦”.
In
particular,
[cf.
the
discussion
of
Theorem
3.11,
(ii),
preceding
the
statement
of
(Ind3)]
multiplication
of
the
data
considered
in
Theorem
3.11,
(ii),
(b),
(c),
by
roots
of
unity
must
be
“identified”
with
the
identity
automorphism.
Put
another
way,
this
data
of
Theo-
rem
3.11,
(ii),
(b),
(c),
may
only
be
considered
up
to
multiplication
by
roots
of
unity.
Thus,
for
instance,
it
only
makes
sense
to
consider
orbits
of
this
data
rel-
ative
to
multiplication
by
roots
of
unity
[i.e.,
as
opposed
to
specific
elements
within
such
orbits].
This
does
not
cause
any
problems
in
the
case
of
the
theta
values
considered
in
Theorem
3.11,
(ii),
(b),
precisely
because
the
theory
developed
so
far
was
formulated
precisely
in
such
a
way
as
to
be
invariant
with
respect
to
such
indeterminacies
[i.e.,
multiplication
of
the
theta
values
by
2l-th
roots
of
unity
—
cf.
the
left-hand
portion
of
Fig.
3.4
below].
In
the
case
of
the
number
fields
[i.e.,
copies
of
F
mod
]
considered
in
Theorem
3.11,
(ii),
(c),
the
resulting
indeterminacies
do
not
cause
any
problems
precisely
because,
in
the
theory
of
the
present
series
of
papers,
ultimately
one
is
only
interested
in
the
global
Frobenioids
[i.e.,
copies
of
”
and
“F
mod
”
and
their
realifications]
associated
to
these
number
fields
by
“F
MOD
means
of
constructions
that
only
involve
·
local
data,
together
with
·
the
entire
set
—
i.e.,
which,
unlike
specific
elements
of
this
set,
is
stabilized
by
multiplication
by
roots
of
unity
of
the
number
field
[cf.
the
left-hand
portion
of
Fig.
3.5
below]
—
constituted
by
the
number
field
under
consideration
[cf.
the
constructions
of
Example
3.6,
(i),
(ii);
the
discussion
of
Remark
3.9.2].
In
this
context,
we
recall
from
the
discussion
of
Remark
2.3.3,
(vi),
that
the
operation
of
forgetting
the
labels
of
the
various
Frobenius-like
“•’s”
also
gives
rise
to
various
indeterminacies
in
the
cyclotomic
rigidity
isomorphisms
applied
in
the
log-Kummer
correspondence.
On
the
other
hand,
in
the
case
of
the
theta
values
considered
in
Theorem
3.11,
(ii),
(b),
we
recall
from
this
discussion
of
172
SHINICHI
MOCHIZUKI
Remark
2.3.3,
(vi),
that
such
indeterminacies
are
in
fact
trivial
[cf.
the
right-hand
portion
of
Fig.
3.4
below].
In
the
case
of
the
number
fields
[i.e.,
copies
of
F
mod
]
considered
in
Theorem
3.11,
(ii),
(c),
we
recall
from
this
discussion
of
Remark
2.3.3,
(vi),
that
such
cyclotomic
rigidity
isomorphism
indeterminacies
amount
to
a
possible
indeterminacy
of
multiplication
by
±1
on
copies
of
the
multiplicative
×
[cf.
the
right-hand
portion
of
Fig.
3.5
below],
i.e.,
indeterminacies
group
F
mod
which
do
not
cause
any
problems,
again,
precisely
as
a
consequence
of
the
fact
that
such
indeterminacies
stabilize
the
entire
set
[i.e.,
as
opposed
to
specific
elements
of
this
set]
constituted
by
the
number
field
under
consideration.
Finally,
in
this
context,
we
observe
[cf.
the
discussion
at
the
beginning
of
Remark
2.3.3,
(viii)]
that,
in
the
case
of
the
various
local
data
at
v
∈
V
non
that
appears
in
Theorem
3.11,
(ii),
(a),
and
gives
rise
to
the
holomorphic
log-shells
that
serve
as
containers
for
the
data
considered
in
Theorem
3.11,
(ii),
(b),
(c),
the
corresponding
cyclotomic
rigidity
isomorphism
indeterminacies
are
in
fact
trivial.
Indeed,
this
triviality
may
be
understood
as
a
consequence
of
the
fact
the
following
observation:
Unlike
the
case
with
the
cyclotomic
rigidity
isomorphisms
that
are
applied
in
the
context
of
the
geometric
containers
[cf.
the
discussion
of
Remark
2.3.3,
(i)]
that
appear
in
the
case
of
the
data
of
Theorem
3.11,
(ii),
(b),
(c),
i.e.,
which
give
rise
to
“vicious
circles”/“loops”
consisting
of
identification
morphisms
that
differ
from
the
usual
natural
identification
by
multiplication
by
elements
of
the
submonoid
I
ord
⊆
±N
≥1
[cf.
the
discussion
of
Remark
2.3.3,
(vi)],
the
cyclotomic
rigidity
isomorphisms
that
are
applied
in
the
context
of
this
local
data
—
even
when
subject
to
the
various
identifications
aris-
ing
from
forgetting
the
labels
of
the
various
Frobenius-like
“•’s”!
—
only
give
rise
to
natural
isomorphisms
between
“geometric”
cyclo-
tomes
arising
from
the
geometric
fundamental
group
and
“arithmetic”
cyclotomes
arising
from
copies
of
the
absolute
Galois
group
of
the
base
[local]
field
[cf.
[AbsTopIII],
Corollary
1.10,
(c);
[AbsTopIII],
Proposition
3.2,
(i),
(ii);
[AbsTopIII],
Remark
3.2.1].
That
is
to
say,
no
“vicious
circles”/“loops”
arise
since
there
is
never
any
confu-
sion
between
such
“geometric”
and
“arithmetic”
cyclotomes.
[A
similar
phenome-
non
may
be
observed
at
v
∈
V
arc
with
regard
to
the
Kummer
structures
considered
in
[IUTchI],
Example
3.4,
(i).]
Thus,
in
summary,
the
various
indeterminacies
that,
a
priori,
might
arise
in
the
context
of
the
portions
of
the
log-Kummer
correspondence
that
appear
in
the
final
portion
of
Theorem
3.11,
(iii),
(c),
(d),
are
in
fact
“invisible”,
i.e.,
they
have
no
substantive
effect
on
the
objects
under
consideration
[cf.
also
the
discussion
of
(ii)
below].
This
is
precisely
the
sense
in
which
the
“compatibility”
stated
in
the
final
portion
of
Theorem
3.11,
(iii),
(c),
(d),
is
to
be
understood.
(ii)
In
the
context
of
the
discussion
of
(i),
we
make
the
following
observation:
the
discussion
in
(i)
of
indeterminacies
that,
a
priori,
might
arise
in
the
context
of
the
portions
of
the
log-Kummer
correspondence
that
appear
in
the
final
portion
of
Theorem
3.11,
(iii),
(c),
(d),
is
complete,
i.e.,
there
are
no
further
possible
indeterminacies
that
might
appear.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
173
Indeed,
this
observation
is
a
consequence
of
the
“general
nonsense”
observation
[cf.,
e.g.,
the
discussion
of
[FrdII],
Definition
2.1,
(ii)]
that,
in
general,
“Kummer
isomorphisms”
are
completely
determined
by
the
following
data:
(a)
isomorphisms
between
the
respective
cyclotomes
under
consideration;
(b)
the
Galois
action
on
roots
of
elements
of
the
monoid
under
considera-
tion.
That
is
to
say,
the
compatibility
of
all
of
the
various
constructions
that
appear
with
the
actions
of
the
relevant
Galois
groups
[or
arithmetic
fundamental
groups]
is
tautological,
so
there
is
no
possibility
that
further
indeterminacies
might
arise
with
respect
to
the
data
of
(b).
On
the
other
hand,
the
effect
of
the
indeterminacies
that
might
arise
with
respect
to
the
data
of
(a)
was
precisely
the
content
of
the
latter
portion
of
the
discussion
of
(i)
[i.e.,
of
the
discussion
of
Remark
2.3.3,
(vi),
(viii)].
(iii)
In
the
context
of
the
discussion
of
(i),
we
observe
that
the
“invisible
indeterminacies”
discussed
in
(i)
in
the
case
of
the
data
considered
in
Theorem
3.11,
(ii),
(b),
(c),
may
be
thought
of
as
a
sort
of
analogue
for
this
data
of
the
indeterminacy
(Ind3)
[cf.
the
discussion
of
the
final
portion
of
Theorem
3.11,
(ii)]
to
which
the
data
of
Theorem
3.11,
(ii),
(a),
is
subject.
By
contrast,
the
multiradiality
and
radial/coric
decoupling
discussed
in
Remarks
2.3.2,
2.3.3
[cf.
also
Theorem
3.11,
(iii),
(c),
(d)]
may
be
understood
as
asserting
precisely
that
the
indeterminacies
(Ind1),
(Ind2)
discussed
in
Theorem
3.11,
(i),
which
act,
essentially,
on
the
data
of
Theorem
3.11,
(ii),
(a),
have
no
effect
on
the
geometric
containers
[cf.
the
discussion
of
Remark
2.3.3,
(i)]
that
underlie
[i.e.,
prior
to
execution
of
the
relevant
evaluation
operations]
the
data
considered
in
Theorem
3.11,
(ii),
(b),
(c).
2
q
j
{1}
(⊆
±N
≥1
)
μ
2l
j=1,...,l
Fig.
3.4:
Invisible
indeterminacies
acting
on
theta
values
×
μ(F
mod
)
×
F
mod
{±1}
(⊆
±N
≥1
)
×
Fig.
3.5:
Invisible
indeterminacies
acting
on
copies
of
F
mod
The
following
result
may
be
thought
of
as
a
relatively
concrete
consequence
of
the
somewhat
abstract
content
of
Theorem
3.11.
Corollary
3.12.
(Log-volume
Estimates
for
Θ-Pilot
Objects)
Suppose
that
we
are
in
the
situation
of
Theorem
3.11.
Write
−
|log(Θ)|
∈
R
{+∞}
for
the
procession-normalized
mono-analytic
log-volume
[i.e.,
where
the
av-
erage
is
taken
over
j
∈
F
l
—
cf.
Remark
3.1.1,
(ii),
(iii),
(iv);
Proposition
3.9,
(i),
(ii);
Theorem
3.11,
(i),
(a)]
of
the
holomorphic
hull
[cf.
Remark
3.9.5,
(i)]
of
the
union
of
the
possible
images
of
a
Θ-pilot
object
[cf.
Definition
3.8,
(i)],
174
SHINICHI
MOCHIZUKI
relative
to
the
relevant
Kummer
isomorphisms
[cf.
Theorem
3.11,
(ii)],
in
the
multiradial
representation
of
Theorem
3.11,
(i),
which
we
regard
as
subject
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
described
in
Theorem
3.11,
(i),
(ii).
Write
−
|log(q)|
∈
R
for
the
procession-normalized
mono-analytic
log-volume
of
the
image
of
a
q-pilot
object
[cf.
Definition
3.8,
(i)],
relative
to
the
relevant
Kummer
isomor-
phisms
[cf.
Theorem
3.11,
(ii)],
in
the
multiradial
representation
of
Theorem
3.11,
(i),
which
we
do
not
regard
as
subject
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
described
in
Theorem
3.11,
(i),
(ii).
Here,
we
recall
the
definition
of
the
symbol
“”
as
the
result
of
identifying
the
labels
“0”
and
“F
l
”
[cf.
[IUTchII],
Corollary
4.10,
(i)].
In
particular,
|log(q)|
>
0
is
easily
computed
in
terms
of
the
various
q-parameters
of
the
elliptic
curve
E
F
[cf.
[IUTchI],
Definition
3.1,
(b)]
at
v
∈
V
bad
(
=
∅).
Then
it
holds
that
−
|log(Θ)|
∈
R,
and
−
|log(Θ)|
≥
−
|log(q)|
—
i.e.,
C
Θ
≥
−1
for
any
real
number
C
Θ
∈
R
such
that
−
|log(Θ)|
≤
C
Θ
·
|log(q)|.
Proof.
We
begin
by
observing
that,
since
|log(q)|
>
0,
we
may
assume
without
loss
of
generality
in
the
remainder
of
the
proof
that
−
|log(Θ)|
<
0
whenever
−
|log(Θ)|
∈
R
[i.e.,
since
an
inequality
−
|log(Θ)|
≥
0
would
imply
that
−
|log(Θ)|
≥
0
>
−|log(q)|].
Now
suppose
that
we
are
in
the
situation
of
Theorem
3.11.
For
n
∈
Z,
write
def
def
n,◦
n,◦
n,◦
Q
U
=
U
j,v
Q
⊆
n,◦
U
Q
=
U
j,v
Q
j∈|F
l
|,v
Q
∈V
Q
j∈|F
l
|,v
Q
∈V
Q
[where
we
observe
that
the
“⊆”
constitutes
a
slight
abuse
of
notation]
for
the
±
Q
=
I
Q
(
S
j+1
;n,◦
D
v
Q
)
[cf.
Theorem
3.11,
(i),
collection
of
subsets
n,◦
U
j,v
Q
⊆
n,◦
U
j,v
Q
(a)]
given
by
the
various
unions,
for
j
∈
|F
l
|
and
v
Q
∈
V
Q
,
of
the
possible
images
of
a
Θ-pilot
object
[cf.
Definition
3.8,
(i)],
relative
to
the
relevant
Kummer
isomorphisms
[cf.
Theorem
3.11,
(ii)],
in
the
multiradial
representation
of
Theorem
3.11,
(i),
which
we
regard
as
subject
to
the
indeterminacies
(Ind1),
(Ind2),
(Ind3)
described
in
Theorem
3.11,
(i),
(ii);
n,◦
n,◦
n,◦
Q
U
=
U
j,v
Q
⊆
n,◦
U
Q
=
U
j,v
Q
def
j∈|F
l
|,v
Q
∈V
Q
j∈|F
l
|,v
Q
∈V
Q
[where
we
observe
that
the
“⊆”
constitutes
a
slight
abuse
of
notation]
for
the
col-
±
Q
Q
S
j+1
;n,◦
lection
of
subsets
n,◦
U
j,v
Q
⊆
n,◦
U
j,v
=
I
(
D
v
Q
)
[cf.
Theorem
3.11,
(i),
(a)]
Q
given
by
the
various
holomorphic
hulls
[cf.
Remark
3.9.5,
(i)]
of
the
subsets
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
175
±
U
j,v
Q
⊆
I
Q
(
S
j+1
;n,◦
D
v
Q
),
relative
to
the
arithmetic
holomorphic
structure
labeled
“n,
◦”.
Here,
we
observe
that
one
concludes
easily
from
the
[easily
verified]
com-
pactness
of
the
1,◦
U
j,v
Q
[where
j
∈
|F
l
|,
v
Q
∈
V
Q
],
together
with
the
definition
of
the
log-volume,
that
the
quantity
−
|log(Θ)|
is
finite,
hence
negative
[by
our
assumption
at
the
beginning
of
the
present
proof!].
In
particular,
we
observe
[cf.
Remark
2.4.2,
(iv),
(v),
(vi);
Remark
3.9.6;
Remark
3.9.7;
the
discussion
of
“IPL”
in
Remark
3.11.1,
(iii)]
that
n,◦
we
may
restrict
our
attention
to
possible
images
of
a
Θ-pilot
object
that
correspond
to
data
[i.e.,
collections
of
regions]
that
may
be
interpreted
as
an
F
-prime-strip.
Now
we
proceed
to
review
precisely
what
is
achieved
by
the
various
portions
of
Theorem
3.11
and,
indeed,
by
the
theory
developed
thus
far
in
the
present
series
of
papers.
This
review
leads
naturally
to
an
interpretation
of
the
theory
that
gives
rise
to
the
inequality
asserted
in
the
statement
of
Corollary
3.12.
For
ease
of
reference,
we
divide
our
discussion
into
steps,
as
follows.
(i)
In
the
following
discussion,
we
concentrate
on
a
single
arrow
—
i.e.,
a
single
Θ
×μ
LGP
-link
±ell
±ell
Θ
×μ
LGP
0,0
1,0
HT
Θ
NF
−→
HT
Θ
NF
—
of
the
[LGP-Gaussian]
log-theta-lattice
under
consideration.
This
arrow
consists
of
the
full
poly-isomorphism
of
F
×μ
-prime-strips
0,0
×μ
F
LGP
∼
→
1,0
F
×μ
[cf.
Definition
3.8,
(ii)].
This
poly-isomorphism
may
be
thought
of
as
consisting
of
a
“unit
portion”
constituted
by
the
associated
[full]
poly-isomorphism
of
F
×μ
-
prime-strips
0,0
×μ
∼
1,0
×μ
F
LGP
→
F
and
a
“value
group
portion”
constituted
by
the
associated
[full]
poly-isomorphism
of
F
-prime-strips
∼
0,0
F
LGP
→
1,0
F
[cf.
Definition
2.4,
(iii)].
This
value
group
portion
of
the
Θ
×μ
LGP
-link
maps
Θ-pilot
Θ
±ell
NF
Θ
±ell
NF
0,0
1,0
to
q-pilot
objects
of
HT
[cf.
Remark
3.8.1].
objects
of
HT
(ii)
Whereas
the
units
of
the
Frobenioids
that
appear
in
the
F
×μ
-prime-strip
0,0
×μ
F
LGP
are
subject
to
Aut
F
×μ
(−)-indeterminacies
[i.e.,
“(Ind1),
(Ind2)”
—
cf.
Theorem
3.11,
(iii),
(a),
(b)],
the
cyclotomes
that
appear
in
the
Kummer
theory
surrounding
the
étale
theta
function
and
κ-coric
functions,
i.e.,
which
give
rise
to
the
“value
group
portion”
0,0
F
LGP
,
are
insulated
from
these
Aut
F
×μ
(−)-
indeterminacies
—
cf.
Theorem
3.11,
(iii),
(c),
(d);
the
discussion
of
Remark
3.11.3,
(iv);
Fig.
3.6
below.
Here,
we
recall
that
in
the
case
of
the
étale
theta
function,
this
follows
from
the
theory
of
§2,
i.e.,
in
essence,
from
the
cyclotomic
rigidity
of
mono-theta
environments,
as
discussed
in
[EtTh].
On
the
other
hand,
in
the
case
of
κ-coric
functions,
this
follows
from
the
algorithms
discussed
in
[IUTchI],
Example
5.1,
(v);
[IUTchI],
Definition
5.2,
(vi),
(viii).
176
SHINICHI
MOCHIZUKI
require
mono-analytic
containers,
Kummer
theory
incompatible
with
(Ind1),
(Ind2)
independent
of
mono-analytic
containers,
Kummer
theory
compatible
with
(Ind1),
(Ind2)
[cf.
Remark
2.3.3]
Θ-related
objects
NF-related
objects
local
LGP-monoids
copies
of
F
mod
[cf.
Proposition
3.4,
(ii)]
[cf.
Proposition
3.7,
(i)]
étale
theta
function,
mono-theta
environments
[cf.
Corollary
2.3]
global
∞
κ-coric,
local
∞
κ-,
∞
κ×-coric
structures
[cf.
Remark
2.3.2]
Fig.
3.6:
Relationship
of
theta-
and
number
field-related
objects
to
mono-analytic
containers
(iii)
In
the
following
discussion,
it
will
be
of
crucial
importance
to
relate
si-
multaneously
both
the
unit
and
the
value
group
portions
of
the
Θ
×μ
LGP
-link(s)
involved
on
the
0-column
[i.e.,
the
vertical
line
indexed
by
0]
of
the
log-theta-lattice
under
consideration
to
the
corresponding
unit
and
value
group
portions
on
the
1-column
[i.e.,
the
vertical
line
indexed
by
1]
of
the
log-theta-lattice
under
con-
sideration.
On
the
other
hand,
if
one
attempts
to
relate
the
unit
portions
via
one
Θ
×μ
LGP
-link
[say,
from
(0,
m)
to
(1,
m)]
and
the
value
group
portions
via
another
-link
[say,
from
(0,
m
)
to
(1,
m
),
for
m
=
m],
then
the
non-commutativity
Θ
×μ
LGP
of
the
log-theta-lattice
renders
it
practically
impossible
to
obtain
conclusions
that
require
one
to
relate
both
the
unit
and
the
value
group
portions
simultaneously
[cf.
the
discussion
of
Remark
3.11.3,
(i),
(ii)].
This
is
precisely
why
we
concentrate
on
a
single
Θ
×μ
LGP
-link
[cf.
(i)].
(iv)
The
issue
discussed
in
(iii)
is
relevant
in
the
context
of
the
present
dis-
cussion
for
the
following
reason.
Ultimately,
we
wish
to
apply
the
bi-coricity
of
the
units
[cf.
Theorem
1.5,
(iii),
(iv)]
in
order
to
compute
the
0-column
Θ-pilot
object
in
terms
of
the
arithmetic
holomorphic
structure
of
the
1-column.
In
order
to
do
this,
one
must
work
with
units
that
are
vertically
once-shifted
[i.e.,
lie
at
(n,
m
−
1)]
relative
to
the
value
group
structures
involved
[i.e.,
which
lie
at
(n,
m)]
—
cf.
the
discussion
of
Remark
3.11.3,
(ii).
The
solution
to
the
problem
of
si-
multaneously
accommodating
these
apparently
contradictory
requirements
—
i.e.,
“vertical
shift”
vs.
“impossibility
of
vertical
shift”
[cf.
(iii)]
—
is
given
precisely
by
working,
on
the
0-column,
with
structures
that
are
invariant
with
respect
to
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
177
vertical
shifts
[i.e.,
“(0,
m)
→
(0,
m
+
1)”]
of
the
log-theta-lattice
[cf.
the
discus-
sion
surrounding
Remark
1.2.2,
(iii),
(a)]
such
as
vertically
coric
structures
[i.e.,
indexed
by
“(n,
◦)”]
that
are
related
to
the
“Frobenius-like”
structures
which
are
not
vertically
coric
by
means
of
the
log-Kummer
correspondences
of
Theorem
3.11,
(ii).
Here,
we
note
that
this
“solution”
may
be
implemented
only
at
the
cost
of
admitting
the
“indeterminacy”
constituted
by
the
upper
semi-compatibility
of
(Ind3).
(v)
Thus,
we
begin
our
computation
of
the
0-column
Θ-pilot
object
in
terms
of
the
arithmetic
holomorphic
structure
of
the
1-column
by
relating
the
units
on
the
0-
and
1-columns
by
means
of
the
unit
portion
0,0
×μ
F
LGP
∼
→
1,0
F
×μ
of
the
Θ
×μ
LGP
-link
from
(0,
0)
to
(1,
0)
[cf.
(i)]
and
then
applying
the
bi-coricity
of
the
units
of
Theorem
1.5,
(iii),
(iv).
In
particular,
the
mono-analytic
log-shell
interpretation
of
this
bi-coricity
given
in
Theorem
1.5,
(iv),
will
be
applied
to
regard
these
mono-analytic
log-shells
as
“multiradial
mono-analytic
containers”
[cf.
the
discussion
of
Remark
1.5.2,
(i),
(ii),
(iii)]
for
the
various
[local
and
global]
value
group
structures
that
constitute
the
Θ-pilot
object
on
the
0-column
—
cf.
Fig.
3.6
above.
[Here,
we
observe
that
the
parallel
treatment
of
“theta-related”
and
“number
field-related”
objects
is
reminiscent
of
the
discussion
of
[IUTchII],
Remark
4.11.2,
(iv).]
That
is
to
say,
we
will
relate
the
various
Frobenioid-theoretic
[i.e.,
“Frobenius-like”
—
cf.
Remark
1.5.4,
(i)]
·
local
units
at
v
∈
V,
·
splitting
monoids
at
v
∈
V
bad
,
and
·
global
Frobenioids
indexed
by
(0,
m),
for
m
∈
Z,
to
the
vertically
coric
[i.e.,
indexed
by
“(0,
◦)”]
versions
of
these
bi-coric
mono-analytic
containers
by
means
of
the
log-Kummer
correspondences
of
Theorem
3.11,
(ii),
(a),
(b),
(c)
—
i.e.,
by
varying
the
“Kum-
mer
input
index”
(0,
m)
along
the
0-column.
(vi)
In
the
context
of
(v),
it
is
useful
to
recall
that
the
log-Kummer
correspon-
dences
of
Theorem
3.11,
(ii),
(b),
(c),
are
obtained
precisely
as
a
consequence
of
the
splittings,
up
to
roots
of
unity,
of
the
relevant
monoids
into
unit
and
value
group
portions
constructed
by
applying
the
Galois
evaluation
operations
dis-
cussed
in
Remarks
2.2.2,
(iii)
[in
the
case
of
Theorem
3.11,
(ii),
(b)],
and
2.3.2
[in
the
case
of
Theorem
3.11,
(ii),
(c)].
Moreover,
we
recall
that
the
Kummer
theory
surrounding
the
local
LGP-monoids
of
Proposition
3.4,
(ii),
depends,
in
an
essential
way,
on
the
theory
of
[IUTchII],
§3
[cf.,
especially,
[IUTchII],
Corollaries
3.5,
3.6],
which,
in
turn,
depends,
in
an
essential
way,
on
the
Kummer
theory
surrounding
mono-theta
environments
established
in
[EtTh].
Thus,
for
instance,
we
recall
that
the
discrete
rigidity
established
in
[EtTh]
is
applied
so
as
to
avoid
working,
in
the
tempered
Frobenioids
that
occur,
with
“
Z-divisors/line
bundles”
[i.e.,
“
Z-completions”
of
Z-modules
of
divisors/line
bundles],
which
are
fundamentally
incompatible
with
conventional
notions
of
divisors/line
bundles,
hence,
in
partic-
ular,
with
mono-theta-theoretic
cyclotomic
rigidity
[cf.
Remark
2.1.1,
(v)].
Also,
we
recall
that
“isomorphism
class
compatibility”
—
i.e.,
in
the
terminology
of
178
SHINICHI
MOCHIZUKI
[EtTh],
“compatibility
with
the
topology
of
the
tempered
fundamental
group”
[cf.
the
discussion
at
the
beginning
of
Remark
2.1.1]
—
allows
one
to
apply
the
Kummer
theory
of
mono-theta
environments
[i.e.,
the
theory
of
[EtTh]]
relative
to
the
ring-theoretic
basepoints
that
occur
on
either
side
of
the
log-link
[cf.
Remarks
2.1.1,
(ii),
and
2.3.3,
(vii);
[IUTchII],
Remark
3.6.4,
(i)],
for
instance,
in
the
context
of
the
log-Kummer
correspondence
for
the
splitting
monoids
of
local
LGP-monoids,
whose
construction
depends,
in
an
essential
way
[cf.
the
theory
of
[IUTchII],
§3,
especially,
[IUTchII],
Corollaries
3.5,
3.6],
on
the
conjugate
syn-
chronization
arising
from
the
F
±
l
-symmetry.
That
is
to
say,
it
is
precisely
by
establishing
this
conjugate
synchronization
arising
from
the
F
±
l
-symmetry
relative
to
these
basepoints
that
occur
on
either
side
of
the
log-link
that
one
is
able
to
conclude
the
crucial
compatibility
of
this
conjugate
synchronization
with
the
log-link
[cf.
Remark
1.3.2].
A
similar
observation
may
be
made
concerning
the
MLF-Galois
pair
approach
to
the
cyclotomic
rigidity
isomorphism
that
is
applied
at
v
∈
V
good
V
non
[cf.
[IUTchII],
Corollary
1.11,
(a);
[IUTchII],
Remark
1.11.1,
(i),
(a);
[IUTchII],
Proposition
4.2,
(i);
[AbsTopIII],
Proposition
3.2,
(iv),
as
well
as
Remark
2.3.3,
(viii),
of
the
present
paper],
which
amounts,
in
essence,
to
computations
involving
the
Galois
cohomology
groups
of
various
subquo-
tients
—
such
as
torsion
subgroups
[i.e.,
roots
of
unity]
and
associated
value
groups
—
of
the
[multiplicative]
module
of
nonzero
elements
of
an
algebraic
closure
of
the
mixed
characteristic
local
field
involved
[cf.
the
proof
of
[AbsAnab],
Proposition
1.2.1,
(vii)]
—
i.e.,
algorithms
that
are
manifestly
compatible
with
the
topology
of
the
profinite
groups
involved
[cf.
the
discussion
of
Remark
2.3.3,
(viii)],
in
the
sense
that
they
do
not
require
one
to
pass
to
Kummer
towers
[cf.
the
discussion
of
[IUTchII],
Remark
3.6.4,
(i)],
which
are
fundamentally
incompatible
with
the
ring
structure
of
the
fields
involved.
Here,
we
note
in
passing
that
the
corresponding
property
for
v
∈
V
arc
[cf.
[IUTchII],
Propo-
sition
4.4,
(i)]
holds
as
a
consequence
of
the
interpretation
discussed
in
[IUTchI],
Remark
3.4.2,
of
Kummer
structures
in
terms
of
co-holomorphicizations.
On
the
other
hand,
the
approaches
to
cyclotomic
rigidity
just
discussed
for
v
∈
V
bad
and
v
∈
V
good
differ
quite
fundamentally
from
the
approach
to
cyclotomic
rigidity
taken
in
the
case
of
[global]
number
fields
in
the
algorithms
described
in
[IUTchI],
Example
5.1,
(v);
[IUTchI],
Definition
5.2,
(vi),
(viii),
which
depend,
in
an
essential
way,
on
the
property
×
=
{1}
Z
Q
>0
—
i.e.,
which
is
fundamentally
incompatible
with
the
topology
of
the
profinite
groups
involved
[cf.
the
discussion
of
Remark
2.3.3,
(vi),
(vii),
(viii)]
in
the
sense
that
it
clearly
cannot
be
obtained
as
some
sort
of
limit
of
corresponding
properties
of
(Z/N
Z)
×
!
Nevertheless,
with
regard
to
uni-/multi-radiality
issues,
this
approach
to
cyclotomic
rigidity
in
the
case
of
the
number
fields
resembles
the
theory
of
mono-theta-theoretic
cyclotomic
rigidity
at
v
∈
V
bad
in
that
it
admits
a
natural
multiradial
formulation
[cf.
Theorem
3.11,
(iii),
(d);
the
discussion
of
Remarks
2.3.2,
3.11.3],
in
sharp
contrast
to
the
essentially
uniradial
nature
of
the
approach
to
cyclotomic
rigidity
via
MLF-Galois
pairs
at
v
∈
V
good
V
non
[cf.
the
discussion
of
[IUTchII],
Remark
1.11.3].
These
observations
are
summarized
in
Fig.
3.7
below.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
179
Finally,
we
recall
that
[one
verifies
immediately
that]
the
various
approaches
to
cyclotomic
rigidity
just
discussed
are
mutually
compatible
in
the
sense
that
they
yield
the
same
cyclotomic
rigidity
isomorphism
in
any
setting
in
which
more
than
one
of
these
approaches
may
be
applied.
Approach
to
cyclotomic
rigidity
Uni-/multi-
radiality
Compatibility
with
F
±
l
-symmetry,
profinite/tempered
topologies,
ring
structures,
log-link
mono-theta
environments
multiradial
compatible
MLF-Galois
pairs,
via
Brauer
groups
uniradial
compatible
number
fields,
via
×
=
{1}
Z
Q
>0
multiradial
incompatible
Fig.
3.7:
Three
approaches
to
cyclotomic
rigidity
(vii)
In
the
context
of
the
discussion
in
the
final
portion
of
(vi),
it
is
of
in-
terest
to
recall
that
the
constructions
underlying
the
crucial
bi-coricity
theory
of
Theorem
1.5,
(iii),
(iv),
depend,
in
an
essential
way,
on
the
conjugate
synchro-
nization
arising
from
the
F
±
l
-symmetry,
which
allows
one
to
relate
the
local
monoids
and
Galois
groups
at
distinct
labels
∈
|F
l
|
to
one
another
in
a
fashion
that
is
simultaneously
compatible
both
with
·
the
vertically
coric
structures
and
Kummer
theory
that
give
rise
to
the
log-Kummer
correspondences
of
Theorem
3.11,
(ii),
and
with
·
the
property
of
distinguishing
[i.e.,
not
identifying]
data
indexed
by
distinct
labels
∈
|F
l
|
—
cf.
the
discussion
of
Remark
1.5.1,
(i),
(ii).
Since,
moreover,
this
crucial
conju-
gate
synchronization
is
fundamentally
incompatible
with
the
F
l
-symmetry,
it
is
necessary
to
work
with
these
two
symmetries
separately,
as
was
done
in
[IUTchI],
§4,
§5,
§6
[cf.
[IUTchII],
Remark
4.7.6].
Here,
it
is
useful
to
recall
that
the
F
l
-
symmetry
also
plays
a
crucial
role,
in
that
it
allows
one
to
“descend
to
F
mod
”
at
the
level
of
absolute
Galois
groups
[cf.
[IUTchII],
Remark
4.7.6].
On
the
other
hand,
both
the
F
±
l
-
and
F
l
-symmetries
share
the
property
of
being
compatible
with
the
vertical
coricity
and
relevant
Kummer
isomorphisms
of
the
0-column
—
cf.
the
log-Kummer
correspondences
of
Theorem
3.11,
(ii),
(b)
[in
the
case
of
the
180
SHINICHI
MOCHIZUKI
F
±
l
-symmetry],
(c)
[in
the
case
of
the
F
l
-symmetry].
Here,
we
recall
that
the
vertically
coric
versions
of
both
the
F
±
l
-
and
the
F
l
-symmetries
depend,
in
an
essential
way,
on
the
arithmetic
holomorphic
structure
of
the
0-column,
hence
give
rise
to
multiradial
structures
via
the
tautological
approach
to
constructing
such
structures
discussed
in
Remark
3.11.2,
(i),
(ii).
(viii)
In
the
context
of
(vii),
it
is
useful
to
recall
that
in
order
to
construct
the
F
±
l
-symmetry,
it
is
necessary
to
make
use
of
global
±-synchronizations
of
various
local
±-indeterminacies.
Since
the
local
tempered
fundamental
groups
at
v
∈
V
bad
do
not
extend
to
a
“global
tempered
fundamental
group”,
these
global
±-
synchronizations
give
rise
to
profinite
conjugacy
indeterminacies
in
the
verti-
cally
coric
construction
of
the
LGP-monoids
[i.e.,
the
theta
values
at
torsion
points]
given
in
[IUTchII],
§2,
which
are
resolved
by
applying
the
theory
of
[IUTchI],
§2
—
cf.
the
discussion
of
[IUTchI],
Remark
6.12.4,
(iii);
[IUTchII],
Remark
4.5.3,
(iii);
[IUTchII],
Remark
4.11.2,
(iii).
(ix)
In
the
context
of
(vii),
it
is
also
useful
to
recall
the
important
role
played,
in
the
theory
of
the
present
series
of
papers,
by
the
various
“copies
of
F
mod
”,
i.e.,
more
concretely,
in
the
form
of
the
various
copies
of
the
global
Frobenioids
”,
“F
mod
”
and
their
realifications.
That
is
to
say,
the
ring
structure
of
“F
MOD
the
global
field
F
mod
allows
one
to
bridge
the
gap
—
i.e.,
furnishes
a
translation
apparatus
—
between
the
multiplicative
structures
constituted
by
the
global
realified
Frobenioids
related
via
the
Θ
×μ
LGP
-link
and
the
additive
representations
of
these
global
Frobenioids
that
arise
from
the
“mono-analytic
containers”
furnished
by
the
mono-analytic
log-shells
[cf.
(v)].
Here,
the
precise
compatibility
of
”
with
the
log-Kummer
correspondence
renders
“F
MOD
”
the
ingredients
for
“F
MOD
×μ
better
suited
to
describing
the
relation
to
the
Θ
LGP
-link
[cf.
Remark
3.10.1,
(ii)].
”
—
i.e.,
which
is
subject
to
“upper
On
the
other
hand,
the
local
portion
of
“F
mod
semi-compatibility”
[cf.
(Ind3)],
hence
only
“approximately
compatible”
with
the
log-Kummer
correspondence
—
renders
it
better
suited
to
explicit
estimates
of
global
arithmetic
degrees,
by
means
of
log-volumes
[cf.
Remark
3.10.1,
(iii)].
(x)
Thus,
one
may
summarize
the
discussion
thus
far
as
follows.
The
theory
of
“Kummer-detachment”
—
cf.
Remarks
1.5.4,
(i);
2.1.1;
3.10.1,
(ii),
(iii)
—
fur-
nished
by
Theorem
3.11,
(ii),
(iii),
allows
one
to
relate
the
Frobenioid-theoretic
[i.e.,
“Frobenius-like”]
structures
that
appear
in
the
domain
[i.e.,
at
(0,
0)]
of
the
Θ
×μ
LGP
-link
[cf.
(i)]
to
the
multiradial
representation
described
in
Theorem
3.11,
(i),
(a),
(b),
(c),
but
only
at
the
cost
of
introducing
the
indeterminacies
(Ind1)
—
which
may
be
thought
of
as
arising
from
the
requirement
of
com-
patibility
with
the
permutation
symmetries
of
the
étale-picture
[cf.
Theorem
3.11,
(i)];
(Ind2)
—
which
may
be
thought
of
as
arising
from
the
requirement
of
com-
patibility
with
the
Aut
F
×μ
(−)-indeterminacies
that
act
on
the
do-
main/codomain
of
the
Θ
×μ
LGP
-link
[cf.
(ii);
Theorem
3.11,
(i),
(iii)],
i.e.,
with
the
horizontal
arrows
of
the
log-theta-lattice;
(Ind3)
—
which
may
be
thought
of
as
arising
from
the
requirement
of
compat-
ibility
with
the
log-Kummer
correspondences
of
Theorem
3.11,
(ii),
i.e.,
with
the
vertical
arrows
of
the
log-theta-lattice.
INTER-UNIVERSAL
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THEORY
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181
The
various
indeterminacies
(Ind1),
(Ind2),
(Ind3)
to
which
the
multiradial
repre-
sentation
is
subject
may
be
thought
of
as
data
that
describes
some
sort
of
“formal
quotient”,
like
the
“fine
moduli
spaces”
that
appear
in
algebraic
geometry.
In
this
context,
the
procession-normalized
mono-analytic
log-volumes
[i.e.,
where
the
average
is
taken
over
j
∈
F
l
]
of
Theorem
3.11,
(i),
(a),
(c),
furnish
a
means
of
constructing
a
sort
of
associated
“coarse
space”
or
“inductive
limit”
[of
the
“inductive
system”
constituted
by
this
“formal
quotient”]
—
i.e.,
in
the
sense
that
[one
verifies
immediately
—
cf.
Proposition
3.9,
(ii)
—
that]
the
re-
sulting
log-volumes
∈
R
are
invariant
with
respect
to
the
indeterminacies
(Ind1),
(Ind2),
and
have
the
effect
of
converting
the
indeterminacy
(Ind3)
into
an
in-
equality
[from
above].
Moreover,
the
log-link
compatibility
of
the
various
log-
volumes
that
appear
[cf.
Proposition
3.9,
(iv);
the
final
portion
of
Theorem
3.11,
(ii)]
ensures
that
these
log-volumes
are
compatible
with
[the
portion
of
the
“formal
quotient”/“inductive
system”
constituted
by]
the
various
arrows
[i.e.,
Kummer
iso-
morphisms
and
log-links]
of
the
log-Kummer
correspondence
of
Theorem
3.11,
(ii).
Here,
we
note
that
the
averages
over
j
∈
F
l
that
appear
in
the
definition
of
the
procession-normalized
volumes
involved
may
be
thought
of
as
a
consequence
of
the
F
l
-symmetry
acting
on
the
labels
of
the
theta
values
that
give
rise
to
the
LGP-monoids
—
cf.
also
the
definition
of
the
symbol
“”
in
[IUTchII],
Corollary
4.10,
(i),
via
the
identification
of
the
symbols
“0”
and
“F
l
”;
the
discussion
of
Remark
3.9.3.
Also,
in
this
context,
it
is
of
interest
to
observe
that
the
various
tensor
products
that
appear
in
the
various
local
mono-analytic
tensor
packets
that
arise
in
the
multiradial
representation
of
Theorem
3.11,
(i),
(a),
have
the
ef-
fect
of
identifying
the
operation
of
“multiplication
by
elements
of
Z”
—
and
hence
also
the
effect
on
log-volumes
of
such
multiplication
operations!
—
at
different
labels
∈
F
l
.
(xi)
For
ease
of
reference,
we
divide
this
step
into
substeps,
as
follows.
(xi-a)
Consider
a
q-pilot
object
at
(1,
0),
which
we
think
of
—
relative
to
the
relevant
copy
of
“F
mod
”
—
in
terms
of
the
holomorphic
log-shells
constructed
at
(1,
0)
[cf.
the
discussion
of
Remark
3.12.2,
(iv),
(v),
below].
Then
the
Θ
×μ
LGP
-
link
from
(0,
0)
to
(1,
0)
may
be
interpreted
as
a
sort
of
gluing
isomorphism
that
relates
the
arithmetic
holomorphic
structure
—
i.e.,
the
“conventional
ring/scheme-theory”
—
at
(1,
0)
to
the
arithmetic
holomorphic
structure
at
(0,
0)
in
such
a
way
that
the
Θ-pilot
object
at
(0,
0)
[thought
of
as
an
object
of
the
relevant
global
realified
Frobenioid]
corresponds
to
the
q-pilot
object
at
(1,
0)
[cf.
(i);
the
discussion
of
Remark
3.12.2,
(ii),
below].
(xi-b)
On
the
other
hand,
the
multiradial
construction
algorithm
of
Theorem
3.11,
which
was
summarized
in
the
discussion
of
(x),
yields
a
construction
of
a
collection
of
possibilities
of
output
data
contained
in
(
0,◦
U
Q
⊇)
∼
0,◦
U
∼
→
1,◦
U
(⊆
1,◦
U
Q
)
—
where
the
isomorphism
“
→
”
arises
from
the
permutation
symmetries
dis-
cussed
in
the
final
portion
of
Theorem
3.11,
(i)
—
that
satisfies
the
input
prime-
strip
link
(IPL)
and
simultaneous
holomorphic
expressibility
(SHE)
prop-
erties
discussed
in
Remark
3.11.1,
(iii),
(iv),
(v)
[cf.
also
the
discussion
of
“possible
182
SHINICHI
MOCHIZUKI
images”
at
the
beginning
of
the
present
proof].
Here,
with
regard
to
(IPL),
we
ob-
serve
that
the
F
-prime-strip
portion
of
the
link/relationship
of
this
collection
of
possibilities
of
output
data
to
the
input
data
(F
×μ
-)prime-strip
[cf.
Re-
mark
3.11.1,
(ii)]
consists
precisely
of
(full
poly-)isomorphisms
of
F
-prime-
strips,
while
the
corresponding
link/relationship
for
F
×μ
-prime-strips
is
some-
what
more
complicated,
as
a
result
of
the
indeterminacies
(Ind1),
(Ind2),
(Ind3).
Also,
in
this
context,
we
observe
that,
although
the
multiradial
construction
algo-
rithm
of
Theorem
3.11
in
fact
involves
the
Θ-pilot
object
at
(0,
0),
in
the
present
discussion
of
Step
(xi),
we
shall
only
be
concerned
with
qualitative
logical
as-
pects/consequences
of
this
construction
algorithm,
i.e.,
with
the
·
input
prime-strip
link
(IPL),
·
simultaneous
holomorphic
expressibility
(SHE),
and
·
algorithmic
parallel
transport
(APT)
properties
discussed
in
Remark
3.11.1,
(iii),
(iv),
(v).
That
is
to
say,
we
shall
take
the
point
of
view
of
“temporarily
forgetting”
—
cf.
the
discussion
of
hidden
internal
structures
(HIS)
in
Remark
3.11.1,
(iv)
—
the
fact
that
the
multiradial
construction
algorithm
of
Theorem
3.11
in
fact
involves
Θ-pilot
objects,
theta
functions/values,
mono-theta
environments.
Alternatively,
in
the
discussion
to
follow,
we
shall,
roughly
speaking,
think
of
the
multiradial
construction
algorithm
of
Theorem
3.11
as
“some”
algorithm
that
transforms
a
certain
type
of
input
data
into
a
certain
type
of
output
data
and,
moreover,
satisfies
certain
properties
(IPL)
and
(SHE).
(xi-c)
Thus,
the
discussion
of
the
(IPL)
and
(SHE)
properties
in
(xi-b)
may
be
summarized
as
follows:
The
multiradial
construction
algorithm
of
Theorem
3.11
yields
a
collection
of
possibilities
of
output
data
in
1,◦
U
(⊆
1,◦
U
Q
)
that
are
linked/related
[cf.
(IPL)],
via
isomorphisms
of
F
-prime-strips,
to
the
representa-
tion
[via
the
log-Kummer
correspondence
in
the
1-column]
of
the
q-pilot
object
at
(1,
0)
on
1,◦
U
Q
,
and,
moreover,
whose
construction
may
be
ex-
pressed
entirely
relative
to
the
arithmetic
holomorphic
structure
in
the
1-column
[cf.
(SHE)].
Here,
we
recall
that,
in
more
concrete
language,
this
“arithmetic
holomorphic
struc-
ture
in
the
1-column”
amounts,
in
essence,
to
the
ring
structure
labeled
“1,
◦”.
Moreover,
by
slightly
enlarging
the
collection
of
possibilities
of
output
data
under
consideration
by
working
with
the
holomorphic
hull
1,◦
U
(⊇
1,◦
U),
we
obtain
output
data
that
is
expressed
—
not
in
terms
of
regions
contained
in
various
tensor
products
of
local
fields
labeled
“1,
◦”
[i.e.,
more
concretely,
various
isomorphs
of
“K
v
”,
for
v
∈
V],
but
rather
—
in
terms
of
localizations
of
arithmetic
vector
bundles
over
certain
local
rings
labeled
“1,
◦”
[i.e.,
more
concretely,
various
iso-
morphs
of
“O
K
v
”,
for
v
∈
V]
—
cf.
the
discussion
of
Remarks
3.9.5,
(vii),
(Ob1),
(Ob2),
(Ob5);
3.12.2,
(v),
below.
Such
an
expression
in
terms
of
“localizations
of
arithmetic
vector
bundles”
is
necessary
in
order
to
render
the
output
data
in
a
form
that
is
comparable
to
the
representation
of
the
q-pilot
object
[i.e.,
which
arises
from
a
certain
arithmetic
line
bundle]
at
(1,
0)
on
1,◦
U
Q
.
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THEORY
III
183
(xi-d)
The
discussion
of
(xi-c)
thus
yields
the
following
conclusion:
The
multiradial
construction
algorithm
of
Theorem
3.11,
followed
by
for-
mation
of
the
holomorphic
hull,
yields
a
collection
of
possibilities
of
out-
put
data
in
1,◦
U
that
are
linked/related
[cf.
(IPL)],
via
isomorphisms
of
F
-prime-strips,
to
the
representation
[via
the
log-Kummer
cor-
respondence
in
the
1-column]
of
the
q-pilot
object
at
(1,
0)
on
1,◦
U
Q
,
and,
moreover,
whose
construction
may
be
expressed
entirely
relative
to
localizations
of
arithmetic
vector
bundles
over
rings
that
arise
in
the
arithmetic
holomorphic
structure
in
the
1-column
[cf.
(SHE)].
Here,
we
observe
that
these
“localizations
of
arithmetic
vector
bundles”
are
[unlike
the
arithmetic
line
bundle
that
gives
rise
to
the
q-pilot
object]
of
rank
>
1.
Moreover,
the
q-pilot
object
is
defined
at
the
level
of
realifications
of
Frobenioids
of
[global]
arithmetic
line
bundles.
Thus,
it
is
only
by
forming
[a
suitable
positive
tensor
power
of]
the
determinant
of
the
localizations
of
arithmetic
vector
bundles
mentioned
in
the
above
display
[cf.
Remark
3.9.5,
(vii),
(Ob3),
(Ob4)]
and
then
applying
the
[suitably
normalized,
with
respect
to
j
∈
|F
l
|]
log-volume
to
various
regions
—
i.e.,
the
region
1,◦
U
and
the
region
that
arises
from
the
representation
of
the
q-pilot
object
at
(1,
0)
on
1,◦
U
Q
—
in
1,◦
U
Q
[cf.
Remark
3.9.5,
(vii),
(Ob3),
(Ob4),
(Ob6),
(Ob7),
(Ob9)],
that
we
are
able
to
obtain
completely
comparable
objects
[cf.
Remarks
3.9.5,
(vii),
(Ob5),
(Ob6),
(Ob7),
(Ob8),
(Ob9);
3.9.5,
(viii),
(ix)],
namely,
def
R
≤−|log(Θ)|
=
{λ
∈
R
|
λ
≤
−|log(Θ)|}
⊆
R;
−
|log(q)|
∈
R
—
where
we
recall
that,
by
definition,
−
|log(Θ)|
is
the
[negative
—
cf.
the
discussion
of
“possible
images”
at
the
beginning
of
the
present
proof]
log-volume
of
1,◦
U
,
while
−
|log(q)|
is
the
log-volume
of
the
region
that
arises
from
the
representation
of
the
q-pilot
object
at
(1,
0)
on
1,◦
U
Q
.
In
this
context,
it
is
useful
to
recall
from
Proposition
3.9,
(iii)
[cf.
also
the
discussion
of
Remarks
3.9.2,
3.9.6],
that
global
arithmetic
degrees
of
objects
of
global
realified
Frobenioids
may
be
interpreted
as
log-volumes
[cf.
also
the
discussion
of
Remarks
1.5.2,
(iii);
3.10.1,
(iv),
as
well
as
of
Remark
3.12.2,
(v),
below].
Finally,
in
this
context,
we
observe
[cf.
the
first
display
of
the
present
(xi-d)]
that
it
is
of
crucial
importance
to
apply
the
log-Kummer
correspondence
in
the
1-column
[cf.
the
discussion
of
log-
Kummer
correspondences
in
Remark
3.9.5,
(vii),
(Ob7),
(Ob8);
Remark
3.9.5,
(viii),
(sQ4);
Remark
3.9.5,
(ix);
the
final
portion
of
Remark
3.9.5,
(x);
the
discussion
of
the
final
portion
of
Remark
3.12.2,
(v),
below],
in
order
to
rectify
the
vertical
shift/mismatch
[cf.
the
discussion
of
(iii),
(iv)
in
the
case
of
the
0-column]
between
the
unit
portion
of
1,0
F
×μ
and
the
log-shells
arising
from
[the
image
via
the
relevant
Kummer
isomorphisms
of]
this
unit
portion,
which
give
rise
to
the
tensor
packets
of
log-shells
that
constitute
1,◦
U
.
(xi-e)
Next,
let
us
recall
that
the
relationship,
i.e.,
that
arises
by
applying
the
log-volume
to
the
pilot-object,
between
the
pilot-object
log-volume
−
|log(q)|
∈
R
and
the
input
data
(F
×μ
-)prime-strip
is
precisely
the
relationship
pre-
scribed/imposed
by
the
arithmetic
holomorphic
structure
in
the
1-column,
i.e.,
via
the
representation
of
the
input
data
(F
×μ
-)prime-strip
on
1,◦
U
relative
184
SHINICHI
MOCHIZUKI
to
this
1-column
arithmetic
holomorphic
structure.
That
is
to
say,
“expressibil-
ity
relative
to
the
arithmetic
holomorphic
structure
in
the
1-column”
[cf.
(SHE)]
amounts
precisely
to
“expressibility
via
operations
that
are
valid/executable/well-defined
even
when
subject
to
the
condition
that
the
pilot-object
log-volume
asso-
ciated
to
the
input
data
(F
×μ
-)prime-strip
[which
is,
of
course,
linked/
related,
via
isomorphisms
of
F
-prime-strips,
to
the
possible
output
data
F
-prime-strips!]
be
equal
to
the
fixed
value
−
|log(q)|
∈
R”.
In
particular,
the
discussion
of
(xi-d)
thus
yields
the
following
conclusion:
The
multiradial
construction
algorithm
of
Theorem
3.11,
followed
by
for-
mation
of
the
holomorphic
hull
and
application
of
the
log-volume,
yields
a
collection
of
possible
log-volumes
of
pilot-object
output
data
R
≤−|log(Θ)|
⊆
R
that
are
linked/related
[cf.
(IPL)],
via
isomorphisms
of
F
-prime-
strips,
to
the
pilot-object
log-volume
−
|log(q)|
∈
R
of
the
input
data
(F
×μ
-)prime-strip
[cf.
(SHE)].
(xi-f)
Thus,
we
conclude
from
(xi-e)
that
the
construction
of
the
subset
R
≤−|log(Θ)|
⊆
R
of
possible
pilot-object
log-volumes
of
output
data
is
subject
to
the
condition
that
this
con-
struction
of
output
data
possibilities
constitutes,
in
particular,
a
construc-
tion
[perhaps
only
up
to
some
sort
of
“approximation”,
as
a
result
of
vari-
ous
indeterminacies]
of
the
pilot-object
log-volume
of
the
input
data
(F
×μ
-)prime-strip,
namely,
−
|log(q)|
∈
R.
The
inclusion
−
|log(q)|
∈
R
≤−|log(Θ)|
,
hence
also
the
inequality
−
|log(q)|
≤
−
|log(Θ)|
∈
R
—
i.e.,
the
conclusion
that
C
Θ
≥
−1
for
any
C
Θ
∈
R
such
that
−
|log(Θ)|
≤
C
Θ
·
|log(q)|
—
in
the
statement
of
Corollary
3.12,
then
follows
formally.
(xi-g)
Thus,
in
summary,
the
multiradial
construction
algorithm
of
Theorem
3.11,
followed
by
formation
of
the
holomorphic
hull
and
application
of
the
log-volume,
yields
two
tautologically
equivalent
ways
to
compute
the
log-volume
of
the
q-pilot
object
at
(1,
0)
—
cf.
Fig.
3.8
below.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
multiradial
representation
at
0-column
(0,
◦)
Kummer-
detach-
ment
via
log-
Kummer
⇑
permutation
symmetry
of
≈
étale-picture
com-
pati-
bly
with
Θ
×μ
LGP
-
link
Θ-pilot
object
in
Θ
±ell
NF-Hodge
theater
at
(0,
0)
multiradial
representation
at
1-column
(1,
◦)
com-
pari-
son
via
(−)
-portion,
(−)
×μ
-portion
≈
×μ
of
Θ
LGP
-link
185
⇓
hol.
hull,
log-
vol.
q-pilot
object
in
Θ
±ell
NF-Hodge
theater
at
(1,
0)
Fig.
3.8:
Two
tautologically
equivalent
ways
to
compute
the
log-volume
of
the
q-pilot
object
at
(1,
0)
(xi-h)
In
this
context,
it
is
useful
to
recall
that
the
above
argument
depends,
in
an
essential
way
[cf.
the
discussion
of
(ii),
(vi)],
on
the
theory
of
[EtTh],
which
does
not
admit
any
evident
generalization
to
the
case
of
N
-th
tensor
powers
of
Θ-pilot
objects,
for
N
≥
2.
That
is
to
say,
the
log-volume
of
such
an
N
-th
tensor
power
of
a
Θ-pilot
object
must
always
be
computed
as
the
result
of
multiplying
the
log-volume
of
the
original
Θ-pilot
object
by
N
—
cf.
Remark
2.1.1,
(iv);
[IUTchII],
Remark
3.6.4,
(iii),
(iv).
In
particular,
although
the
analogue
of
the
above
argument
for
such
an
N
-th
tensor
power
would
lead
to
sharper
inequalities
than
the
inequalities
obtained
here,
it
is
difficult
to
see
how
to
obtain
such
sharper
inequalities
via
a
routine
generalization
of
the
above
argument.
In
fact,
as
we
shall
see
in
[IUTchIV],
these
sharper
inequalities
are
known
to
be
false
[cf.
[IUTchIV],
Remark
2.3.2,
(ii)].
(xii)
In
the
context
of
the
argument
of
(xi),
it
is
useful
to
observe
the
important
role
played
by
the
global
realified
Frobenioids
that
appear
in
the
Θ
×μ
LGP
-link.
That
is
to
say,
since
ultimately
one
is
only
concerned
with
the
computation
of
log-volumes,
it
might
appear,
at
first
glance,
that
it
is
possible
to
dispense
with
the
use
of
such
global
Frobenioids
and
instead
work
only
with
the
various
local
Frobenioids,
for
v
∈
V,
that
are
directly
related
to
the
computation
of
log-volumes.
On
the
other
hand,
observe
that
since
the
isomorphism
of
[local
or
global!]
Frobenioids
arising
from
the
Θ
×μ
LGP
-link
only
preserves
isomorphism
classes
of
objects
of
these
Frobenioids
[cf.
the
discussion
of
Remark
3.6.2,
(i)],
to
work
only
with
local
Frobenioids
means
that
one
must
contend
with
the
indeterminacy
of
not
knowing
whether,
for
instance,
such
a
local
Frobenioid
object
at
some
v
∈
V
non
corresponds
to
a
given
open
submodule
of
the
log-shell
at
v
or
to,
say,
the
p
N
v
-multiple
of
this
submodule,
for
N
∈
Z.
Put
another
way,
one
must
contend
with
the
indeterminacy
arising
from
the
fact
that,
unlike
the
case
with
the
global
Frobenioids
“F
MOD
”,
R
“F
MOD
”,
objects
of
the
various
local
Frobenioids
that
arise
admit
endomorphisms
186
SHINICHI
MOCHIZUKI
which
are
not
automorphisms.
This
indeterminacy
has
the
effect
of
rendering
meaningless
any
attempt
to
perform
a
precise
log-volume
computation
as
in
(xi).
Remark
3.12.1.
(i)
In
[IUTchIV],
we
shall
be
concerned
with
obtaining
more
explicit
upper
bounds
on
−
|log(Θ)|,
i.e.,
estimates
“C
Θ
”
as
in
the
statement
of
Corollary
3.12.
(ii)
It
is
not
difficult
to
verify
that,
for
λ
∈
Q
>0
,
one
may
obtain
a
similar
theory
to
the
theory
developed
in
the
present
series
of
papers
[cf.
the
discussion
of
Remark
3.11.1,
(ii)]
for
“generalized
Θ
×μ
LGP
-links”
of
the
form
q
λ
→
1
2
..
.
2
)
(l
q
—
i.e.,
so
the
theory
developed
in
the
present
series
of
papers
corresponds
to
the
case
of
λ
=
1.
This
sort
of
“generalized
Θ
×μ
LGP
-link”
is
roughly
reminiscent
of
—
but
by
no
means
equivalent
to!
—
the
sort
of
issues
considered
in
the
discussion
of
Remark
2.2.2,
(i).
Here,
we
observe
that
raising
to
the
λ-th
power
on
the
“q
2
2
side”
differs
quite
fundamentally
from
raising
to
the
λ-th
power
on
the
“q
(1
...(l
)
)
side”,
an
issue
that
is
discussed
briefly
[in
the
case
of
λ
=
N
]
in
the
final
portion
of
Step
(xi)
of
the
proof
of
Corollary
3.12.
That
is
to
say,
“generalized
Θ
×μ
LGP
-links”
as
in
the
above
display
differ
fundamentally
both
from
the
situation
of
Remark
2.2.2,
(i),
and
the
situation
discussed
in
the
final
portion
of
Step
(xi)
of
the
proof
of
Corollary
3.12
in
that
the
theory
of
the
first
power
of
the
étale
theta
function
is
left
unchanged
[i.e.,
relative
to
the
theory
developed
in
the
present
series
of
papers]
—
cf.
the
discussion
of
Remark
2.2.2,
(i);
Step
(xi)
of
the
proof
of
Corollary
3.12.
At
any
rate,
in
the
case
of
“generalized
Θ
×μ
LGP
-links”
as
in
the
above
display,
one
may
apply
the
same
arguments
as
the
arguments
used
to
prove
Corollary
3.12
to
conclude
the
inequality
C
Θ
≥
−λ
—
i.e.,
which
is
sharper,
for
λ
<
1,
than
the
inequality
obtained
in
Corollary
3.12
in
the
case
of
λ
=
1.
In
fact,
however,
such
sharper
inequalities
will
not
be
of
interest
to
us,
since,
in
[IUTchIV],
our
estimates
for
the
upper
bound
C
Θ
will
be
sufficiently
rough
as
to
be
unaffected
by
adding
a
constant
of
absolute
value
≤
1.
(iii)
In
the
context
of
the
discussion
of
(ii)
above,
it
is
of
interest
to
note
that
the
multiradial
theory
of
mono-theta-theoretic
cyclotomic
rigidity,
and,
in
particular,
the
theory
of
the
first
power
of
the
étale
theta
function,
may
be
regarded
as
a
theory
that
concerns
a
sort
of
“canonical
profinite
volume”
on
the
elliptic
curves
under
consideration
associated
to
the
first
power
of
the
am-
ple
line
bundle
corresponding
to
the
étale
theta
function.
This
point
of
view
is
also
of
interest
in
the
context
of
the
discussion
of
various
approaches
to
cyclotomic
rigidity
summarized
in
Fig.
3.7
[cf.
also
the
discussion
of
Remark
2.3.3].
Indeed,
×
=
{1}”,
which
plays
a
key
role
in
the
multi-
Z
the
elementary
fact
“Q
>0
radial
algorithms
for
cyclotomic
rigidity
isomorphisms
in
the
number
field
case
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
187
[cf.
[IUTchI],
Example
5.1,
(v),
as
well
as
the
discussion
of
Remarks
2.3.2,
2.3.3
of
the
present
paper],
may
be
regarded
as
an
immediate
consequence
of
an
easy
interpretation
of
the
product
formula
in
terms
of
the
geometry
of
the
domain
in
the
archimedean
completion
of
the
number
field
Q
determined
by
the
inequality
“≤
1”,
i.e.,
a
domain
which
may
be
thought
of
as
a
sort
of
concrete
geometric
representation
of
a
“canonical
unit
of
volume”
of
the
number
field
Q.
Remark
3.12.2.
(i)
One
of
the
main
themes
of
the
present
series
of
papers
is
the
issue
of
dismantling
the
two
underlying
combinatorial
dimensions
of
a
number
field
—
cf.
Remarks
1.2.2,
(vi),
of
the
present
paper,
as
well
as
[IUTchI],
Remarks
3.9.3,
6.12.3,
6.12.6;
[IUTchII],
Remarks
4.7.5,
4.7.6,
4.11.2,
4.11.3,
4.11.4.
The
principle
examples
of
this
topic
may
be
summarized
as
follows:
(a)
splittings
of
various
monoids
into
unit
and
value
group
portions;
(b)
separating
the
“F
l
”
arising
from
the
l-torsion
points
of
the
elliptic
curve
—
which
may
be
thought
of
as
a
sort
of
“finite
approximation”
of
Z!
—
into
a
[multiplicative]
F
l
-symmetry
—
which
may
also
be
thought
of
as
corresponding
to
the
global
arithmetic
portion
of
the
arithmetic
funda-
mental
groups
involved
—
and
a(n)
[additive]
F
±
l
-symmetry
—
which
may
also
be
thought
of
as
corresponding
to
the
geometric
portion
of
the
arithmetic
fundamental
groups
involved;
(c)
separating
the
ring
structures
of
the
various
global
number
fields
that
appear
into
their
respective
underlying
additive
structures
—
which
may
be
related
directly
to
the
various
log-shells
that
appear
—
and
their
respective
underlying
multiplicative
structures
—
which
may
be
related
directly
to
the
various
Frobenioids
that
appear.
From
the
point
of
view
of
Theorem
3.11,
example
(a)
may
be
seen
in
the
“non-
interference”
properties
that
underlie
the
log-Kummer
correspondences
of
Theorem
3.11,
(ii),
(b),
(c),
as
well
as
in
the
Θ
×μ
LGP
-link
compatibility
properties
discussed
in
Theorem
3.11,
(ii),
(c),
(d).
(ii)
On
the
other
hand,
another
important
theme
of
the
present
§3
consists
of
the
issue
of
“reassembling”
these
two
dismantled
combinatorial
dimensions
by
means
of
the
multiradial
mono-analytic
containers
furnished
by
the
mono-
analytic
log-shells
—
cf.
Fig.
3.6
—
i.e.,
of
exhibiting
the
extent
to
which
these
two
dismantled
combinatorial
dimensions
cannot
be
separated
from
one
another,
at
least
in
the
case
of
the
Θ-pilot
object,
by
describing
the
“structure
of
the
intertwining”
between
these
two
dimensions
that
existed
prior
to
their
separa-
tion.
From
this
point
of
view,
one
may
think
of
the
multiradial
representations
discussed
in
Theorem
3.11,
(i)
[cf.
also
Theorem
3.11,
(ii),
(iii)],
as
the
final
output
of
this
“reassembling
procedure”
for
Θ-pilot
objects.
From
the
point
of
view
of
example
(a)
of
the
discussion
of
(i),
this
“reassembling
procedure”
allows
one
to
compute/estimate
the
value
group
portions
of
various
monoids
of
arithmetic
interest
in
terms
of
the
unit
group
portions
of
these
monoids.
It
is
precisely
these
estimates
that
give
rise
to
the
inequality
obtained
in
Corollary
3.12.
That
is
to
say,
from
the
point
of
view
of
dismantling/reassembling
the
intertwining
between
188
SHINICHI
MOCHIZUKI
value
group
and
unit
group
portions,
the
argument
of
the
proof
of
Corollary
3.12
may
be
summarized
as
follows:
(a
itw
)
When
considered
from
the
point
of
view
of
log-volumes
of
Θ-pilot
and
q-pilot
objects,
the
correspondence
of
the
Θ
×μ
LGP
-link
[i.e.,
that
sends
Θ-pilot
objects
to
q-pilot
objects]
may
seem
a
bit
“mysterious”
or
even,
at
first
glance,
“self-contradictory”
to
some
readers.
(b
itw
)
On
the
other
hand,
this
correspondence
of
the
Θ
×μ
LGP
-link
is
made
possi-
ble
by
the
fact
that
one
works
with
Θ-pilot
or
q-pilot
objects
in
terms
of
“sufficiently
weakened
data”
[namely,
the
F
×μ
-prime-strips
that
appear
in
the
definition
of
the
Θ
×μ
LGP
-link],
i.e.,
data
that
is
“sufficiently
weak”
that
one
can
no
longer
distinguish
between
Θ-pilot
and
q-pilot
ob-
jects.
(c
itw
)
Thus,
if
one
thinks
of
the
F
×μ
-prime-strips
that
appear
in
the
domain
×μ
and
codomain
of
the
Θ
×μ
-prime-
LGP
-link
as
a
“single
abstract
F
strip”
that
is
regarded/only
known
up
to
isomorphism,
then
the
issue
of
which
log-volume
such
an
abstract
F
×μ
-prime-strip
corresponds
to
[cf.
(a
itw
)]
is
precisely
the
issue
of
“which
intertwining
between
value
group
and
unit
group
portions”
one
considers,
i.e.,
the
issue
of
“which
arithmetic
holomorphic
structure”
[of
the
arithmetic
holomorphic
struc-
tures
that
appear
in
the
domain
and
codomain
of
the
Θ
×μ
LGP
-link]
that
one
works
in.
Put
another
way,
it
is
essentially
a
tautological
consequence
of
the
fact
that
these
two
arithmetic
holomorphic
structures
in
the
domain
and
codomain
of
the
Θ
×μ
LGP
-link
are
distinguished
from
one
another
that
×μ
the
Θ
LGP
-link
yields
a
situation
in
which
both
the
Θ-intertwining
[i.e.,
the
intertwining
associated
to
the
Θ-pilot
object
in
the
domain
of
the
Θ
×μ
LGP
-link]
and
the
q-intertwining
[i.e.,
the
intertwining
associated
to
the
q-pilot
object
in
the
codomain
of
the
Θ
×μ
LGP
-link]
are
simultaneously
valid,
i.e.,
q-intertwining
holds
∧
Θ-intertwining
holds
—
cf.
the
discussion
of
the
“distinct
labels
approach”
in
Remark
3.11.1,
(vii).
(d
itw
)
On
the
other
hand,
from
the
point
of
view
of
the
analogy
between
multiradiality
and
the
classical
theory
of
parallel
transport
via
con-
nections
[cf.
[IUTchII],
Remark
1.7.1],
the
multiradial
representa-
tion
of
Theorem
3.11
[cf.
also
the
discussion
of
Remark
3.11.1,
especially
Remark
3.11.1,
(ii),
(iii)]
asserts
that,
up
to
the
relatively
mild
“mon-
odromy”
constituted
by
the
indeterminacies
(Ind1),
(Ind2),
(Ind3),
one
may
“parallel
transport”
or
“confuse”
the
Θ-pilot
object
in
the
do-
main
of
the
Θ
×μ
LGP
-link,
i.e.,
the
Θ-pilot
object
represented
relative
to
its
“native
intertwining/arithmetic
holomorphic
structure”,
with
the
Θ-pilot
object
represented
relative
to
the
“alien
intertwining/arithmetic
holomor-
phic
structure”
in
the
codomain
of
the
Θ
×μ
LGP
-link.
(e
itw
)
In
particular,
one
may
fix
the
arithmetic
holomorphic
structure
of
the
codomain
of
the
Θ
×μ
LGP
-link,
i.e.,
the
“native
intertwining/arithmetic
holo-
morphic
structure”
associated
to
the
q-pilot
object
in
the
codomain
of
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
189
itw
Θ
×μ
)
and
working
up
to
the
indeter-
LGP
-link,
and
then,
by
applying
(d
minacies
(Ind1),
(Ind2),
(Ind3)
[cf.
also
the
subtleties
discussed
in
(iv),
(v)
below;
Remark
3.9.5,
(vii),
(viii),
(ix)],
construct
the
“native
in-
tertwining/arithmetic
holomorphic
structure”
associated
to
the
Θ-pilot
object
in
the
domain
of
the
Θ
×μ
LGP
-link
as
a
mathematical
structure
that
is
intrinsically
associated
to
the
underlying
structure
of
—
hence,
in
particular,
simultaneously
with/without
invalidating
the
conditions
imposed
by
—
the
“native
intertwining/arithmetic
holomorphic
structure”
associated
to
the
q-pilot
object
in
the
codomain
of
the
Θ
×μ
LGP
-link
[cf.
the
discussion
of
Remark
3.11.1,
especially
Remark
3.11.1,
(ii),
(iii)].
In-
deed,
this
point
of
view
is
precisely
the
point
of
view
that
is
taken
in
the
proof
of
Corollary
3.12
[cf.,
especially,
Step
(xi)].
(f
itw
)
One
way
of
summarizing
the
situation
described
in
(e
itw
)
is
in
terms
of
logical
relations
as
follows.
The
multiradial
representation
of
The-
orem
3.11
[cf.
also
the
discussion
of
Remark
3.11.1]
may
be
thought
of
[cf.
the
first
“=⇒”
of
the
following
display]
as
an
algorithm
for
construct-
ing,
up
to
suitable
indeterminacies
[cf.
the
discussion
of
(e
itw
)],
the
“Θ-intertwining”
as
a
mathematical
structure
that
is
intrinsically
as-
sociated
to
the
underlying
structure
of
—
hence,
in
particular,
simulta-
neously
with/without
invalidating
[cf.
the
logical
relator
“AND”,
i.e.,
“∧”]
the
conditions
imposed
by
—
the
“q-intertwining”,
while
holding
the
“single
abstract
F
×μ
-prime-strip”
of
the
discussion
of
(b
itw
),
(c
itw
)
fixed,
i.e.,
in
symbols:
q-itw.
=⇒
q-itw.
∧
Θ-itw./indets.
=⇒
Θ-itw./indets.
—
where
the
second
“=⇒”
of
the
above
display
is
purely
formal;
“itw.”
and
“/indets.”
are
to
be
understood,
respectively
as
abbreviations
for
“in-
tertwining
holds”
and
“up
to
suitable
indeterminacies”.
Here,
we
observe
that
the
“∧”
of
the
above
display
may
be
regarded
as
the
“image”
of,
hence,
in
particular,
as
a
consequence
of,
the
“∧”
in
the
display
of
(c
itw
),
via
the
various
(sub)quotient
operations
discussed
in
Remark
3.9.5,
(viii),
i.e.,
whose
subtle
compatibility
properties
allow
one
to
conclude
the
“∧”
of
the
above
display
from
the
“∧”
in
the
display
of
(c
itw
).
Thus,
at
the
level
of
logical
relations,
the
q-intertwining,
hence
also
the
log-volume
of
the
q-pilot
ob-
ject
in
the
codomain
of
the
Θ
×μ
LGP
-link,
may
be
thought
of
as
a
special
case
of
the
Θ-intertwining,
i.e.,
at
a
more
concrete
level,
of
the
log-volume
of
the
Θ-pilot
object
in
the
domain
of
the
Θ
×μ
LGP
-link,
regarded
up
to
suitable
indeterminacies.
Corollary
3.12
then
follows,
essentially
formally.
Alternatively,
from
the
point
of
view
of
“[very
rough!]
toy
models”,
i.e.,
whose
goal
lies
solely
in
representing
certain
overall
qualitative
aspects
of
a
situation,
one
may
think
of
the
discussion
of
(a
itw
)
∼
(f
itw
)
given
above
in
the
following
terms:
190
SHINICHI
MOCHIZUKI
(a
toy
)
Consider
two
distinct
copies
q
R
and
Θ
R
of
the
topological
field
of
real
numbers
R,
equipped
with
labels
“q”
and
“Θ”,
together
with
an
abstract
symbol
“∗”
and
assignments
λ
q
:
∗
→
q
(−h)
∈
q
R,
λ
Θ
:
∗
→
Θ
(−2h)
∈
Θ
R,
—
where,
in
the
present
discussion,
we
shall
write
“
q
(−)”,
“
Θ
(−)”
to
denote
the
respective
elements/subsets
of
q
R,
Θ
R
determined
by
an
ele-
ment/subset
“(−)”
of
R;
h
∈
R
>0
is
a
positive
real
number
that
we
are
interested
in
bounding
from
above.
If
one
forgets
the
distinct
labels
“q”
and
“Θ”,
then
these
two
assignments
λ
q
,
λ
Θ
are
mutually
incompatible
and
cannot
be
considered
simultaneously,
i.e.,
they
contradict
one
another
[in
the
sense
that
R
−h
=
−2h
∈
R].
(b
toy
)
One
aspect
of
the
situation
of
(a
toy
)
that
renders
the
simultaneous
con-
sideration
of
the
two
assignments
λ
q
,
λ
Θ
valid
—
i.e.,
at
the
level
of
logical
relations,
∗
→
q
(−h)
∈
q
R
∧
∗
→
Θ
(−2h)
∈
Θ
R
—
is
the
use
of
the
abstract
symbol
“∗”,
i.e.,
which
is,
a
priori,
entirely
unrelated
to
any
copies
of
R
[such
as
q
R,
Θ
R].
(c
toy
)
The
other
aspect
of
the
situation
of
(a
toy
)
that
renders
the
simultaneous
consideration
of
the
two
assignments
λ
q
,
λ
Θ
valid—
i.e.,
at
the
level
of
logical
relations,
∗
→
q
(−h)
∈
q
R
∧
∗
→
Θ
(−2h)
∈
Θ
R
—
is
the
use
of
the
distinct
labels
“q”,
“Θ”
for
the
copies
of
R
that
appear
in
the
assignments
λ
q
,
λ
Θ
.
(d
toy
)
Now
let
us
consider
an
alternative
approach
to
constructing
the
assign-
ment
λ
Θ
:
We
construct
λ
Θ
as
the
“assignment
with
indeterminacies”
Θ
R
≤−2h+
⊆
Θ
R
λ
Ind
Θ
:
∗
→
def
—
where
R
≤−2h+
=
{x
∈
R
|
x
≤
−2h+}
⊆
R;
∈
R
>0
is
some
positive
number.
(e
toy
)
Now
suppose
that
one
verifies
that
one
may
construct
the
“assignment
toy
)
as
a
mathematical
structure
that
is
with
indeterminacies”
λ
Ind
Θ
of
(d
intrinsically
associated
to
the
underlying
structure
of
the
assignment
λ
q
—
hence,
in
particular,
simultaneously
with/without
invalidating
the
conditions
imposed
by
—
the
assignment
λ
q
,
even
if
one
forgets
the
labels
“q”,
“Θ”
that
were
appended
to
copies
of
R,
i.e.,
even
if
one
identifies
q
R,
Θ
R,
in
the
usual
way,
with
R
[cf.
the
properties
(IPL),
(SHE)
of
Remark
3.11.1,
(iii)].
That
is
to
say,
we
suppose
that
one
can
show
that
the
assignments
determined,
respectively,
by
λ
q
,
λ
Ind
Θ
,
by
identifying
copies
of
R,
namely,
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
∗
→
−h
∈
R,
191
∗
→
R
≤−2h+
⊆
R
—
where
the
latter
assignment
may
be
considered
as
the
assignment
that
maps
∗
to
“some
[undetermined]
element
∈
R
≤−2h+
”
—
are
such
that
one
may
construct
the
latter
assignment
as
a
mathematical
structure
that
is
intrinsically
associated
to
—
hence,
in
particular,
simultaneously
with/without
invalidating
the
conditions
imposed
by
—
the
former
as-
signment.
Here,
we
note
that
it
is
not
particularly
relevant
that
“R
≤−2h+
”
arose
as
some
sort
of
“perturbation
via
indeterminacies
of
2h”
[cf.
the
property
(HIS)
of
Remark
3.11.1,
(iv)].
(f
toy
)
The
discussion
of
(e
toy
)
may
be
summarized
at
the
level
of
logical
relations
[cf.
the
displays
of
(b
toy
),
(c
toy
)]
as
follows:
∗
→
−h
=⇒
∗
→
−h
∧
∗
→
R
≤−2h+
=⇒
∗
→
R
≤−2h+
—
that
is
to
say,
“∗
→
−h”
may
be
regarded
as
a
special
case
of
“∗
→
R
≤−2h+
”,
which,
in
turn,
may
be
regarded
as
a
“version
with
indeterminacies”
of
“∗
→
−2h”.
One
then
concludes
formally
that
−h
∈
R
≤−2h+
and
hence
that
−h
≤
−2h
+
,
i.e.,
h
≤
—
that
is
to
say,
the
desired
upper
bound
on
h.
(iii)
One
fundamental
aspect
of
the
theory
that
renders
possible
the
“reassem-
bling
procedure”
discussed
in
(ii)
[cf.
the
discussion
of
Step
(iv)
of
the
proof
of
Corollary
3.12]
is
the
“juggling
of
,
”
[cf.
the
discussion
of
Remark
1.2.2,
(vi)]
effected
by
the
log-links,
i.e.,
the
vertical
arrows
of
the
log-theta-lattice.
This
“juggling
of
,
”
may
be
thought
of
as
a
sort
of
combinatorial
way
of
represent-
ing
the
arithmetic
holomorphic
structure
associated
to
a
vertical
line
of
the
log-theta-lattice.
Indeed,
at
archimedean
primes,
this
juggling
amounts
essentially
to
multiplication
by
±i,
which
is
a
well-known
method
[cf.
the
notion
of
an
“al-
most
complex
structure”!]
for
representing
holomorphic
structures
in
the
classical
theory
of
differential
manifolds.
On
the
other
hand,
it
is
important
to
recall
in
this
context
that
this
“juggling
of
,
”
is
precisely
what
gives
rise
to
the
up-
per
semi-compatibility
indeterminacy
(Ind3)
[cf.
Proposition
3.5,
(ii);
Remark
3.10.1,
(i)].
(iv)
In
the
context
of
the
discussion
of
(ii),
(iii),
it
is
of
interest
to
compare,
in
the
cases
of
the
0-
and
1-columns
of
the
log-theta-lattice,
the
way
in
which
the
theory
of
log-Kummer
correspondences
associated
to
a
vertical
column
of
the
log-theta-lattice
is
applied
in
the
proof
of
Corollary
3.12,
especially
in
Steps
(x)
and
(xi).
We
begin
by
observing
that
the
vertical
column
[i.e.,
0-
or
1-column]
under
consideration
may
be
depicted
[“horizontally”!]
in
the
fashion
of
the
diagram
of
192
SHINICHI
MOCHIZUKI
the
third
display
of
Proposition
1.3,
(iv)
•
0
...
→
...
•
→
•
↓
→
•
→
...
...
◦
—
where
the
“•
0
”
in
the
first
line
of
the
diagram
denotes
the
portion
with
vertical
coordinate
0
[i.e.,
the
portion
at
(0,
0)
or
(1,
0)]
of
the
vertical
column
under
consid-
eration.
As
discussed
in
Step
(iii)
of
the
proof
of
Corollary
3.12,
since
the
Θ
×μ
LGP
-link
is
fundamentally
incompatible
with
the
distinct
arithmetic
holomorphic
structures
—
i.e.,
ring
structures
—
that
exist
in
the
0-
and
1-columns,
one
is
obliged
to
work
with
the
Frobenius-like
versions
of
the
unit
group
and
value
group
portions
of
monoids
arising
from
“•
0
”
in
the
definition
of
the
Θ
×μ
LGP
-link
precisely
×μ
in
order
to
avoid
the
need
to
contend,
in
the
definition
of
the
Θ
LGP
-link,
with
the
issue
of
describing
the
“structure
of
the
intertwining”
[cf.
the
discussion
of
(ii)]
between
these
unit
group
and
value
group
portions
determined
by
the
distinct
arithmetic
holomorphic
structures
—
i.e.,
ring
structures
—
that
exist
in
the
0-
and
1-columns.
On
the
other
hand,
one
is
also
obliged
to
work
with
the
étale-like
“◦”
versions
of
various
objects
since
it
is
precisely
these
vertically
coric
versions
that
allow
one
to
access,
i.e.,
by
serving
as
containers
[cf.
the
discussion
of
(ii)]
for,
the
other
“•’s”
in
the
vertical
column
under
consideration.
That
is
to
say,
although
the
various
Kummer
isomorphisms
that
relate
various
portions
of
the
Frobenius-like
“•
0
”
to
the
corresponding
portions
of
the
étale-like
“◦”
may
at
first
give
the
impression
that
either
“•
0
”
or
“◦”
is
superfluous
or
unnecessary
in
the
theory,
in
fact
both
“•
0
”
and
“◦”
play
an
essential
and
by
no
means
superfluous
role
in
the
theory
of
the
vertical
columns
of
the
log-theta-lattice.
This
aspect
of
the
theory
is
essentially
the
same
in
the
case
of
both
the
0-
and
the
1-columns.
The
log-link
compatibility
of
the
various
log-volumes
that
appear
[cf.
the
discussion
of
Step
(x)
of
the
proof
of
Corollary
3.12;
Proposition
3.9,
(iv);
the
final
portion
of
Theorem
3.11,
(ii)]
is
another
aspect
of
the
theory
that
is
essentially
the
same
in
the
case
of
both
the
0-
and
the
1-columns.
Also,
although
the
discussion
of
the
“non-interference”
properties
that
underlie
the
log-Kummer
correspondences
of
Theorem
3.11,
(ii),
(b),
(c),
was
only
given
expicitly,
in
effect,
in
the
case
of
the
0-column,
i.e.,
concerning
Θ-pilot
objects,
entirely
similar
“non-
interference”
properties
hold
for
q-pilot
objects.
[Indeed,
this
may
be
seen,
for
instance,
by
applying
the
same
arguments
as
the
arguments
that
were
applied
in
the
case
of
Θ-pilot
objects,
or,
for
instance,
by
specializing
the
non-interference
properties
obtained
for
Θ-pilot
objects
to
the
index
“j
=
1”
as
in
the
discussion
of
“pivotal
distributions”
in
[IUTchI],
Example
5.4,
(vii).]
These
similarities
between
the
0-
and
1-columns
are
summarized
in
the
upper
portion
of
Fig.
3.9
below.
(v)
In
the
discussion
of
(iv),
we
highlighted
various
similarities
between
the
0-
and
1-columns
of
the
log-theta-lattice
in
the
context
of
Steps
(x),
(xi)
of
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
193
proof
of
Corollary
3.12.
By
contrast,
one
significant
difference
between
the
theory
of
log-Kummer
correspondences
in
the
0-
and
1-columns
is
the
lack
of
analogues
for
q-pilot
objects
of
the
crucial
multiradiality
properties
summarized
in
Theorem
3.11,
(iii),
(c)
Aspect
of
the
theory
0-column/
Θ-pilot
objects
1-column/
q-pilot
objects
essential
role
of
both
“•
0
”
and
“◦”
similar
similar
log-link
compatibility
of
log-volumes
similar
similar
similar
similar
multiradiality
properties
of
Θ-/q-pilot
objects
hold
do
not
hold
treatment
of
log-shells/
unit
group
portions
used
as
mono-analytic
containers
for
regions
tautological
documenting
device
for
logarithmic
relationship
betw.
ring
structures
resulting
indeterminacies
acting
on
log-shells
(Ind1),
(Ind2),
(Ind3)
absorbed
by
applying
holomorphic
hulls,
log-volumes
“non-interference”
properties
of
log-Kummer
correspondences
Fig.
3.9:
Similarities
and
differences,
in
the
context
of
the
Θ
×μ
LGP
-link,
between
the
0-
and
1-columns
of
the
log-theta-lattice
—
i.e.,
in
effect,
the
lack
of
an
analogue
for
the
q-pilot
objects
of
the
theory
of
rigidity
properties
developed
in
[EtTh]
[cf.
the
discussion
of
Remark
2.2.2,
(i)].
Another
significant
difference
between
the
theory
of
log-Kummer
correspondences
194
SHINICHI
MOCHIZUKI
in
the
0-
and
1-columns
lies
in
the
way
in
which
the
associated
vertically
coric
holo-
morphic
log-shells
[cf.
Proposition
1.2,
(ix)]
are
treated
in
their
relationship
to
the
unit
group
portions
of
monoids
that
occur
in
the
various
“•’s”
of
the
log-Kummer
correspondence.
That
is
to
say,
in
the
case
of
the
0-column,
these
log-shells
are
used
as
containers
[cf.
the
discussion
of
(ii)]
for
the
various
regions
[i.e.,
sub-
sets]
arising
from
these
unit
group
portions
via
various
composites
of
arrows
in
the
log-Kummer
correspondence.
This
approach
has
the
advantage
of
admitting
an
in-
terpretation
—
i.e.,
in
terms
of
subsets
of
mono-analytic
log-shells
—
that
makes
sense
even
relative
to
the
distinct
arithmetic
holomorphic
structures
that
appear
in
the
1-column
of
the
log-theta-lattice
[cf.
Remark
3.11.1].
On
the
other
hand,
it
has
the
drawback
that
it
gives
rise
to
the
upper
semi-compatibility
indeterminacy
(Ind3)
discussed
in
the
final
portion
of
Theorem
3.11,
(ii).
By
contrast,
in
the
case
of
the
1-column,
since
the
associated
arithmetic
holomor-
phic
structure
is
held
fixed
and
regarded
[cf.
the
discussion
of
Step
(xi)
of
the
proof
of
Corollary
3.12]
as
the
standard
with
respect
to
which
constructions
arising
from
the
0-column
are
to
be
computed,
there
is
no
need
[i.e.,
in
the
case
of
the
1-column]
to
require
that
the
constructions
applied
admit
mono-analytic
interpretations.
That
is
to
say,
in
the
case
of
the
1-column,
the
various
unit
group
portions
of
monoids
at
the
various
“•’s”
simply
serve
as
a
means
of
documenting
the
“log-
arithmic”
relationship
[cf.
the
definition
of
the
log-link
given
in
Definition
1.1,
(i),
(ii)!]
between
the
ring
structures
in
the
domain
and
codomain
of
the
log-link.
These
ring
structures
give
rise
to
the
local
copies
of
sets
of
integral
elements
“O”
with
respect
to
which
the
“mod”
versions
[cf.
Example
3.6,
(ii)]
of
categories
of
arithmetic
line
bundles
are
defined
at
the
various
“•’s”.
Since
the
objects
of
these
categories
of
arithmetic
line
bundles
are
not
equipped
with
local
trivializations
at
the
various
v
∈
V
[cf.
the
discussion
of
isomorphism
classes
of
objects
of
Frobenioids
in
Remark
3.6.2,
(i)],
regions
in
log-shells
may
only
be
related
to
such
categories
of
arithmetic
line
bundles
at
the
expense
of
allowing
for
an
indeterminacy
with
respect
to
“O
×
”-multiples
at
each
v
∈
V.
It
is
precisely
this
indeterminacy
that
necessitates
the
introduction,
in
Step
(xi)
of
the
proof
of
Corollary
3.12,
of
holomorphic
hulls,
i.e.,
which
have
the
effect
of
absorbing
this
indeterminacy
[cf.
the
discussion
of
Remark
3.9.5,
(vii),
(viii),
(ix),
(x),
for
more
details].
Finally,
in
Step
(xi)
of
the
proof
of
Corollary
3.12,
the
indeterminacy
in
the
specification
of
a
particular
member
of
the
collection
of
ring
structures
just
discussed
—
i.e.,
arising
from
the
choice
of
a
particular
composite
of
arrows
in
the
log-Kummer
correspondence
that
is
used
to
specify
a
particular
ring
structure
among
its
various
“logarithmic
conjugates”
—
is
absorbed
by
passing
to
log-volumes
—
i.e.,
by
applying
the
log-link
compatibility
[cf.
(iv)]
of
the
various
log-volumes
associated
to
these
ring
structures
[cf.
the
discussion
of
Remark
3.9.5,
(vii),
(viii),
(ix),
(x),
for
more
details].
Thus,
unlike
the
case
of
the
0-column,
where
the
mono-
analytic
interpretation
via
regions
of
mono-analytic
log-shells
gives
rise
only
to
upper
bounds
on
log-volumes,
the
approach
just
discussed
in
the
case
of
the
1-
column
—
i.e.,
which
makes
essential
use
of
the
ring
structures
that
are
available
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
195
as
a
consequence
of
the
fact
that
the
arithmetic
holomorphic
structure
is
held
fixed
—
gives
rise
to
precise
equalities
[i.e.,
not
just
inequalities!]
concerning
log-volumes.
These
differences
between
the
0-
and
1-columns
are
summarized
in
the
lower
portion
of
Fig.
3.9.
Remark
3.12.3.
(i)
Let
S
be
a
hyperbolic
Riemann
surface
of
finite
type
of
genus
g
S
with
r
S
def
punctures.
Write
χ
S
=
−(2g
S
−
2
+
r
S
)
for
the
Euler
characteristic
of
S
and
dμ
S
for
the
Kähler
metric
on
S
[i.e.,
the
(1,
1)-form]
determined
by
the
Poincaré
metric
on
the
upper
half-plane.
Recall
the
analogy
discussed
in
[IUTchI],
Remark
4.3.3,
between
the
theory
of
log-shells,
which
plays
a
key
role
in
the
theory
developed
in
the
present
series
of
papers,
and
the
classical
metric
geometry
of
hyperbolic
Riemann
surfaces.
Then,
relative
to
this
analogy,
the
inequality
obtained
in
Corollary
3.12
may
be
regarded
as
corresponding
to
the
inequality
χ
S
=
−
S
dμ
S
<
0
—
i.e.,
in
essence,
a
statement
of
the
hyperbolicity
of
S
—
arising
from
the
clas-
sical
Gauss-Bonnet
formula,
together
with
the
positivity
of
dμ
S
.
Relative
to
the
analogy
between
real
analytic
Kähler
metrics
and
ordinary
Frobenius
liftings
discussed
in
[pOrd],
Introduction,
§2
[cf.
also
the
discussion
of
[pTeich],
Introduc-
tion,
§0],
the
local
property
constituted
by
this
positivity
of
dμ
S
may
be
thought
of
as
corresponding
to
the
[local
property
constituted
by
the]
Kodaira-Spencer
iso-
morphism
of
an
indigenous
bundle
—
i.e.,
which
gives
rise
to
the
ordinarity
of
the
corresponding
Frobenius
lifting
on
the
ordinary
locus
—
in
the
p-adic
theory.
As
discussed
in
[AbsTopIII],
§I5,
these
properties
of
indigenous
bundles
in
the
p-adic
theory
may
be
thought
of
as
corresponding,
in
the
theory
of
log-shells,
to
the
“max-
imal
incompatibility”
between
the
various
Kummer
isomorphisms
and
the
corically
constructed
data
of
the
Frobenius-picture
of
Proposition
1.2,
(x).
On
the
other
hand,
it
is
just
this
“maximal
incompatibility”
that
gives
rise
to
the
“upper
semi-
commutativity”
discussed
in
Remark
1.2.2,
(iii),
i.e.,
[from
the
point
of
view
of
the
theory
of
the
present
§3]
the
upper
semi-compatibility
indeterminacy
(Ind3)
of
Theorem
3.11,
(ii),
that
underlies
the
inequality
of
Corollary
3.12
[cf.
Step
(x)
of
the
proof
of
Corollary
3.12].
(ii)
The
“metric
aspect”
of
Corollary
3.12
discussed
in
(i)
is
reminiscent
of
the
analogy
between
the
theory
of
the
present
series
of
papers
and
classical
complex
Teichmüller
theory
[cf.
the
discussion
of
[IUTchI],
Remark
3.9.3]
in
the
following
sense:
Just
as
classical
complex
Teichmüller
theory
is
concerned
with
relating
distinct
holomorphic
structures
in
a
sufficiently
canonical
way
as
to
min-
imize
the
resulting
conformality
distortion,
the
canonical
nature
of
the
algorithms
discussed
in
Theorem
3.11
for
relating
alien
arithmetic
holomorphic
structures
[cf.
Remark
3.11.1]
gives
rise
to
a
relatively
strong
estimate
of
the
[log-]volume
distortion
[cf.
Corollary
3.12]
re-
sulting
from
such
a
deformation
of
the
arithmetic
holomorphic
structure.
196
SHINICHI
MOCHIZUKI
Remark
3.12.4.
In
light
of
the
discussion
of
Remark
3.12.3,
it
is
of
interest
to
reconsider
the
analogy
between
the
theory
of
the
present
series
of
papers
and
the
p-adic
Teichmüller
theory
of
[pOrd],
[pTeich],
in
the
context
of
Theorem
3.11,
Corollary
3.12.
(i)
First,
we
observe
that
the
splitting
monoids
at
v
∈
V
bad
[cf.
Theorem
3.11,
(i),
(b);
Theorem
3.11,
(ii),
(b)]
may
be
regarded
as
analogous
to
the
canoni-
cal
coordinates
of
p-adic
Teichmüller
theory
[cf.,
e.g.,
[pTeich],
Introduction,
§0.9]
that
are
constructed
over
the
ordinary
locus
of
a
canonical
curve.
In
particular,
it
is
natural
to
regard
the
bad
primes
∈
V
bad
as
corresponding
to
the
ordinary
locus
of
a
canonical
curve
and
the
good
primes
∈
V
good
as
corresponding
to
the
supersingular
locus
of
a
canonical
curve.
This
point
of
view
is
reminiscent
of
the
discussion
of
[IUTchII],
Remark
4.11.4,
(iii).
(ii)
On
the
other
hand,
the
bi-coric
mono-analytic
log-shells
—
i.e.,
the
various
local
“O
×μ
”
—
that
appear
in
the
tensor
packets
of
Theorem
3.11,
(i),
(a);
Theorem
3.11,
(ii),
(a),
may
be
thought
of
as
corresponding
to
the
[multi-
plicative!]
Teichmüller
representatives
associated
to
the
various
Witt
rings
that
appear
in
p-adic
Teichmüller
theory.
Within
a
fixed
arithmetic
holomorphic
structure,
these
mono-analytic
log-shells
arise
from
“local
holomorphic
units”
—
i.e.,
“O
×
”
—
which
are
subject
to
the
F
±
l
-symmetry.
These
“local
holomor-
phic
units”
may
be
thought
of
as
corresponding
to
the
positive
characteristic
ring
structures
on
[the
positive
characteristic
reductions
of]
Teichmüller
repre-
sentatives.
Here,
the
uniradial,
i.e.,
“non-multiradial”,
nature
of
these
“local
holomorphic
units”
[cf.
the
discussion
of
[IUTchII],
Remark
4.7.4,
(ii);
[IUTchII],
Figs.
4.1,
4.2]
may
be
regarded
as
corresponding
to
the
mixed
characteristic
nature
of
Witt
rings,
i.e.,
the
incompatibility
of
Teichmüller
representatives
with
the
additive
structure
of
Witt
rings.
(iii)
The
set
F
l
of
l
“theta
value
labels”,
which
plays
an
important
role
in
the
theory
of
the
present
series
of
papers,
may
be
thought
of
as
corresponding
to
the
“factor
of
p”
that
appears
in
the
“mod
p/p
2
portion”,
i.e.,
the
gap
separating
the
“mod
p”
and
“mod
p
2
”
portions,
of
the
rings
of
Witt
vectors
that
occur
in
the
p-adic
theory.
From
this
point
of
view,
one
may
think
of
the
procession-normalized
volumes
obtained
by
taking
averages
over
j
∈
F
l
[cf.
Corollary
3.12]
as
corre-
sponding
to
the
operation
of
dividing
by
p
to
relate
the
“mod
p/p
2
portion”
of
the
Witt
vectors
to
the
“mod
p
portion”
of
the
Witt
vectors
[i.e.,
the
characteristic
p
theory].
In
this
context,
the
multiradial
representation
of
Theorem
3.11,
(i),
by
means
of
mono-analytic
log-shells
labeled
by
elements
of
F
l
may
be
thought
of
as
corresponding
to
the
derivative
of
the
canonical
Frobenius
lifting
on
a
canon-
ical
curve
in
the
p-adic
theory
[cf.
the
discussion
of
[AbsTopIII],
§I5]
in
the
sense
that
this
multiradial
representation
may
be
regarded
as
a
sort
of
comparison
of
the
canonical
splitting
monoids
discussed
in
(i)
to
the
“absolute
constants”
[cf.
the
discussion
of
(ii)]
constituted
by
the
bi-coric
mono-analytic
log-shells.
This
“absolute
comparison”
is
precisely
what
results
in
the
indeterminacies
(Ind1),
(Ind2)
of
Theorem
3.11,
(i).
(iv)
In
the
context
of
the
discussion
of
(iii),
we
note
that
the
set
of
labels
F
l
may,
alternatively,
be
thought
of
as
corresponding
to
the
infinitesimal
moduli
of
the
positive
characteristic
curve
under
consideration
in
the
p-adic
theory
[cf.
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
197
discussion
of
[IUTchII],
Remark
4.11.4,
(iii),
(d)].
That
is
to
say,
the
“deformation
dimension”
constituted
by
the
horizontal
dimension
of
the
log-theta-lattice
in
the
theory
of
the
present
series
of
papers
or
by
the
deformations
modulo
various
powers
of
p
in
the
p-adic
theory
[cf.
Remark
1.4.1,
(iii);
Fig.
1.3]
is
highly
canonical
in
nature,
hence
may
be
thought
of
as
being
equipped
with
a
natural
isomorphism
to
the
“absolute
moduli”
—
i.e.,
so
to
speak,
the
“moduli
over
F
1
”
—
of
the
given
number
field
equipped
with
an
elliptic
curve,
in
the
theory
of
the
present
series
of
papers,
or
of
the
given
positive
characteristic
hyperbolic
curve
equipped
with
a
nilpotent
ordinary
indigenous
bundle,
in
p-adic
Teichmüller
theory.
Inter-universal
Teichmüller
theory
p-adic
Teichmüller
theory
splitting
monoids
at
v
∈
V
bad
canonical
coordinates
on
the
ordinary
locus
bad
primes
∈
V
bad
ordinary
locus
of
a
can.
curve
good
primes
∈
V
good
supersing.
locus
of
a
can.
curve
mono-analytic
log-shells
“O
×μ
”
[multiplicative!]
Teich.
reps.
uniradial
“local
hol.
units
O
×
”
subject
to
F
±
l
-symmetry
pos.
char.
ring
structures
on
[pos.
char.
reductions
of]
Teich.
reps.
set
of
“theta
value
labels”
F
l
factor
p
in
mod
p/p
portion
of
Witt
vectors
multiradial
rep.
via
F
l
-labeled
mono-analytic
log-shells
[cf.
(Ind1),
(Ind2),
(Ind3)]
derivative
of
the
canonical
Frobenius
lifting
set
of
“theta
value
labels”
F
l
implicit
“absolute
moduli/F
1
”
inequality
arising
from
upper
semi-compatibility
[cf.
(Ind3)]
inequality
arising
from
interference
between
Frobenius
conjugates
2
Fig.
3.10:
The
analogy
between
inter-universal
Teichmüller
theory
and
p-adic
Teichmüller
theory
198
SHINICHI
MOCHIZUKI
(v)
Let
A
be
the
ring
of
Witt
vectors
of
a
perfect
field
k
of
positive
characteristic
p;
X
a
smooth,
proper
hyperbolic
curve
over
A
of
genus
g
X
which
is
canonical
in
the
p-adic
formal
scheme
associated
the
sense
of
p-adic
Teichmüller
theory;
X
⊆
X
the
ordinary
locus
of
X.
Write
ω
X
for
the
canonical
bundle
of
to
X;
U
k
def
X
k
=
X
×
A
k.
Then
when
[cf.
the
discussion
of
(iii)]
one
computes
the
derivative
→
U
on
U
,
one
must
contend
with
of
the
canonical
Frobenius
lifting
Φ
:
U
“interference
phenomena”
between
the
various
copies
of
some
positive
characteristic
algebraic
geometry
set-up
—
i.e.,
at
a
more
concrete
level,
the
various
Frobenius
n
conjugates
“t
p
”
[where
t
is
a
local
coordinate
on
X
k
]
associated
to
various
n
∈
N
≥1
.
In
particular,
this
derivative
only
yields
[upon
dividing
by
p]
an
inclusion
[i.e.,
not
an
isomorphism!]
of
line
bundles
ω
X
k
→
Φ
∗
ω
X
k
—
also
known
as
the
“[square]
Hasse
invariant”
[cf.
[pOrd],
Chapter
II,
Propo-
sition
2.6;
the
discussion
of
“generalities
on
ordinary
Frobenius
liftings”
given
in
[pOrd],
Chapter
III,
§1].
Thus,
at
the
level
of
global
degrees
of
line
bundles,
we
obtain
an
inequality
[i.e.,
not
an
equality!]
(1
−
p)(2g
X
−
2)
≤
0
—
which
may
be
thought
of
as
being,
in
essence,
a
statement
of
the
hyperbolicity
of
X
[cf.
the
inequality
of
the
display
of
Remark
3.12.3,
(i)].
Since
the
“Frobenius
n
conjugate
dimension”
[i.e.,
the
“n”
that
appears
in
“t
p
”]
in
the
p-adic
theory
corresponds
to
the
vertical
dimension
of
the
log-theta-lattice
in
the
theory
of
the
present
series
of
papers
[cf.
Remark
1.4.1,
(iii);
Fig.
1.3],
we
thus
see
that
the
inequality
of
the
above
display
in
the
p-adic
case
arises
from
circumstances
that
are
entirely
analogous
to
the
circumstances
—
i.e.,
the
upper
semi-compatibility
indeterminacy
(Ind3)
of
Theorem
3.11,
(ii)
—
that
underlie
the
inequality
of
Corollary
3.12
[cf.
Step
(x)
of
the
proof
of
Corollary
3.12;
the
discussion
of
Remark
3.12.3,
(i)].
(vi)
The
analogies
of
the
above
discussion
are
summarized
in
Fig.
3.10
above.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
III
199
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